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0704.0649
%!PS-Adobe-2.0 %%Creator: dvips(k) 5.95a Copyright 2005 Radical Eye Software %%Title: arXiv:0704.0649v2 [math.RA] 18 Apr 2007 %%Pages: 58 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentFonts: XYATIP10 XYBTIP10 XYDASH10 %%DocumentPaperSizes: Letter %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -z -R -K1 derweyzel-1-v2.dvi -o %DVIPSParameters: dpi=600, comments removed %DVIPSSource: TeX output 2007.04.19:0113 %%BeginProcSet: tex.pro 0 0 /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round 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eq{userdict begin md length 10 add md maxlength ge{/md md dup length 20 add dict copy def}if end md begin /letter{}N/note{}N/legal{}N/od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{ itransform lineto}}{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{closepath}}pathforall newpath counttomark array astore/gc xdf pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack} if}N/txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{ noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N/cp{pop pop showpage pm restore}N end}if}if}N/normalscale{ Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale }if 0 setgray}N/psfts{S 65781.76 div N}N/startTexFig{/psf$SavedState save N userdict maxlength dict begin/magscale true def normalscale currentpoint TR/psf$ury psfts/psf$urx psfts/psf$lly psfts/psf$llx psfts /psf$y psfts/psf$x psfts currentpoint/psf$cy X/psf$cx X/psf$sx psf$x psf$urx psf$llx sub div N/psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR/showpage{}N/erasepage{}N/setpagedevice{pop}N/copypage{}N/p 3 def @MacSetUp}N/doclip{psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath moveto}N/endTexFig{end psf$SavedState restore}N /@beginspecial{SDict begin/SpecialSave save N gsave normalscale currentpoint TR @SpecialDefaults count/ocount X/dcount countdictstack N} N/@setspecial{CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR}{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury lineto closepath clip}if/showpage{}N/erasepage{}N /setpagedevice{pop}N/copypage{}N newpath}N/@endspecial{count ocount sub{ pop}repeat countdictstack dcount sub{end}repeat grestore SpecialSave restore end}N/@defspecial{SDict begin}N/@fedspecial{end}B/li{lineto}B /rl{rlineto}B/rc{rcurveto}B/np{/SaveX currentpoint/SaveY X N 1 setlinecap newpath}N/st{stroke SaveX SaveY moveto}N/fil{fill SaveX SaveY moveto}N/ellipse{/endangle X/startangle X/yrad X/xrad X/savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end %%EndProcSet %%BeginProcSet: hps.pro 0 0 /HPSdict 20 dict dup begin/braindeaddistill 50 def/rfch{dup length 1 sub 1 exch getinterval}bind def/splituri{dup(#)search{exch pop}{()exch} ifelse dup(file:)anchorsearch{pop exch pop 3 -1 roll pop false}{pop 3 -1 roll exch pop true}ifelse}bind def/lookuptarget{exch rfch dup /TargetAnchors where{pop TargetAnchors dup 3 -1 roll known{exch get true }{pop(target unknown:)print == false}ifelse}{pop pop (target dictionary unknown\012)print false}ifelse}bind def/savecount 0 def/stackstopped{count counttomark sub/savecount exch store stopped count savecount sub 1 sub dup 0 gt{{exch pop}repeat}{pop}ifelse}bind def /tempstring 256 string def/targetvalidate{1 index dup length 255 gt exch dup(/)search{pop pop pop exch pop true exch}{pop}ifelse cvn tempstring cvs token pop pop length 0 ne or not}bind def/targetdump-hook where{pop} {/targetdump-hook{dup mark exch gsave initmat setmatrix{{mark/Dest 4 2 roll targetvalidate{aload pop exch pop/Page 3 1 roll/View exch[exch /FitH exch]/DEST pdfmark}{cleartomark}ifelse}forall}stackstopped pop grestore}bind def}ifelse/baseurl{mark exch 1 dict dup 3 -1 roll/Base exch put/URI exch/DOCVIEW{pdfmark}stackstopped pop}bind def /externalhack systemdict/PDF known def/oldstyle true def/initmat matrix currentmatrix def/actiondict 2 dict dup/Subtype/URI put def /weblinkhandler{dup 3 1 roll mark 4 1 roll/Title 4 1 roll splituri 3 -1 roll dup length 0 gt{cvn/Dest exch 4 2 roll}{pop}ifelse{externalhack{ /HTTPFile exch}{actiondict dup 3 -1 roll/URI exch put/Action exch} ifelse}{externalhack{/HTTPFile exch}{/File exch/Action/GoToR}ifelse} ifelse counttomark 2 sub -1 roll aload pop/Rect 4 1 roll/Border 3 1 roll /Color exch oldstyle{/LNK}{/Subtype/Link/ANN}ifelse gsave initmat setmatrix{pdfmark}stackstopped grestore}bind def/externalhandler where{ pop}{/externalhandler{2 copy{weblinkhandler}exec{/externalhack externalhack not store 2 copy{weblinkhandler}exec{/externalhack externalhack not store/oldstyle false store 2 copy{weblinkhandler}exec{ (WARNING: external refs disabled\012)print/externalhandler{pop pop}bind store externalhandler}{pop pop}ifelse}{pop pop/externalhack externalhack not store}ifelse}{pop pop/externalhandler{weblinkhandler pop}bind store} ifelse}bind def}ifelse/pdfmnew{dup type/stringtype eq{externalhandler}{ exch dup rfch exch 3 -1 roll lookuptarget{mark 4 1 roll/Title 4 1 roll aload pop exch pop/Page 3 1 roll/View exch[exch/FitH exch]5 -1 roll aload pop/Rect 4 1 roll/Border 3 1 roll/Color exch/LNK gsave initmat setmatrix pdfmark grestore}{pop pop}ifelse}ifelse}bind def/pdfmold{dup type/stringtype eq{externalhandler}{exch dup rfch exch 3 -1 roll lookuptarget{mark 4 1 roll/Title 4 1 roll aload pop exch pop/Page 3 1 roll/View exch[exch/FitH exch]5 -1 roll aload pop pop 0 3 getinterval /Rect 3 1 roll/Border exch/LNK gsave initmat setmatrix pdfmark grestore} {pop pop}ifelse}ifelse}bind def/pdfm where{pop}{/pdfm /currentdistillerparams where{pop currentdistillerparams dup /CoreDistVersion known{/CoreDistVersion get}{0}ifelse dup braindeaddistill le{(WARNING: switching to old pdfm because version =) print ==/pdfmold}{pop/pdfmnew}ifelse load}{/pdfmark where{pop{dup type /stringtype eq{externalhandler}{2 copy mark 3 1 roll{pdfmnew} stackstopped{2 copy mark 3 1 roll{pdfmold}stackstopped{ (WARNING: pdfm disabled\012)print/pdfm{pop pop}store}{ (WARNING: new pdfm failed, switching to old pdfm\012)print/pdfm/pdfmold load store}ifelse}{/pdfm/pdfmnew load store}ifelse pop pop}ifelse}}{{ pop pop}}ifelse}ifelse bind def}ifelse end def %%EndProcSet TeXDict begin @defspecial /DvipsToPDF { 72.27 mul Resolution div } def /PDFToDvips { 72.27 div Resolution mul } def /HyperBorder { 1 PDFToDvips } def /H.V {pdf@hoff pdf@voff null} def /H.B {/Rect[pdf@llx pdf@lly pdf@urx pdf@ury]} def /H.S { currentpoint HyperBorder add /pdf@lly exch def dup DvipsToPDF /pdf@hoff exch def HyperBorder sub /pdf@llx exch def } def /H.L { 2 sub dup /HyperBasePt exch def PDFToDvips /HyperBaseDvips exch def currentpoint HyperBaseDvips sub /pdf@ury exch def /pdf@urx exch def } def /H.A { H.L currentpoint exch pop vsize 72 sub exch DvipsToPDF HyperBasePt sub sub /pdf@voff exch def } def /H.R { currentpoint HyperBorder sub /pdf@ury exch def HyperBorder add /pdf@urx exch def currentpoint exch pop vsize 72 sub exch DvipsToPDF sub /pdf@voff exch def } def systemdict /pdfmark known { userdict /?pdfmark systemdict /exec get put }{ userdict /?pdfmark systemdict /pop get put userdict /pdfmark systemdict /cleartomark get put } ifelse @fedspecial end %%BeginFont: XYDASH10 %!PS-AdobeFont-1.1: XYDASH10 001.104 %%CreationDate: 1997 Jul 20 21:19:18 %%RevisionDate: 1997 Aug 28 05:34:12 %%RevisionDate: 1997 Sep 18 10:23:31 % % XYDASH10: line segments for Xy-pic at 10 point % % Original Metafont design Copyright (C) 1991-1997 Kristoffer H. Rose. % PostScript adaptation Copyright (C) 1994-1997 Ross Moore. % Hinting and ATM compatibility Copyright (C) 1997 Y&Y, Inc. % % This file is part of the Xy-pic macro package. % Xy-pic Copyright (c) 1991-1997 Kristoffer H. Rose <krisrose@brics.dk> % % The Xy-pic macro package is free software; you can redistribute it % and/or modify it under the terms of the GNU General Public License % as published by the Free Software Foundation; either version 2 % of the License, or (at your option) any later version. % % The Xy-pic macro package is distributed in the hope that it will % be useful, but WITHOUT ANY WARRANTY; without even the implied % warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. % See the GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this macro package; if not, write to the % Free Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA. 11 dict begin /FontInfo 9 dict dup begin /version (001.104) readonly def /Notice (Copyright (C) 1996, 1997 Ross Moore and Y&Y, Inc.) readonly def /FullName (XYDASH10) readonly def /FamilyName (XYDASH) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -300 def /UnderlineThickness 150 def end readonly def /FontName /XYDASH10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /d0 put dup 1 /d1 put dup 61 /d61 put dup 69 /d69 put dup 70 /d70 put dup 121 /d121 put dup 122 /d122 put readonly def /FontBBox{-40 -520 503 520}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: XYBTIP10 %!PS-AdobeFont-1.1: XYBTIP10 001.104 %%CreationDate: 1997 Jul 20 21:19:18 %%RevisionDate: 1997 Sep 14 19:58:47 % % XYBTIP10: lower arrow tips for Xy-pic at 10 point "technical style". % % Original Metafont design Copyright (C) 1991-1997 Kristoffer H. Rose. % PostScript adaptation Copyright (C) 1994-1997 Ross Moore. % Hinting and ATM compatibility Copyright (C) 1997 Y&Y, Inc. % % This file is part of the Xy-pic macro package. % Xy-pic Copyright (c) 1991-1997 Kristoffer H. Rose <krisrose@brics.dk> % % The Xy-pic macro package is free software; you can redistribute it % and/or modify it under the terms of the GNU General Public License % as published by the Free Software Foundation; either version 2 % of the License, or (at your option) any later version. % % The Xy-pic macro package is distributed in the hope that it will % be useful, but WITHOUT ANY WARRANTY; without even the implied % warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. % See the GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this macro package; if not, write to the % Free Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA. 11 dict begin /FontInfo 9 dict dup begin /version (001.104) readonly def /Notice (Copyright (C) 1996, 1997 Ross Moore and Y&Y, Inc.) readonly def /FullName (XYBTIP10) readonly def /FamilyName (XYBTIP) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -276 def /UnderlineThickness 138 def end readonly def /FontName /XYBTIP10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /d0 put dup 15 /d15 put dup 30 /d30 put dup 34 /d34 put dup 47 /d47 put dup 56 /d56 put dup 60 /d60 put dup 63 /d63 put dup 64 /d64 put dup 79 /d79 put dup 94 /d94 put dup 111 /d111 put dup 127 /d127 put readonly def /FontBBox{-542 -542 542 542}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: XYATIP10 %!PS-AdobeFont-1.1: XYATIP10 001.104 %%CreationDate: 1997 Jul 20 21:19:17 %%RevisionDate: 1997 Sep 14 19:58:47 % % XYATIP10: upper arrow tips for Xy-pic at 10 point "technical style". % % Original Metafont design Copyright (C) 1991-1997 Kristoffer H. Rose. % PostScript adaptation Copyright (C) 1994-1997 Ross Moore. % Hinting and ATM compatibility Copyright (C) 1997 Y&Y, Inc. % % This file is part of the Xy-pic macro package. % Xy-pic Copyright (c) 1991-1997 Kristoffer H. Rose <krisrose@brics.dk> % % The Xy-pic macro package is free software; you can redistribute it % and/or modify it under the terms of the GNU General Public License % as published by the Free Software Foundation; either version 2 % of the License, or (at your option) any later version. % % The Xy-pic macro package is distributed in the hope that it will % be useful, but WITHOUT ANY WARRANTY; without even the implied % warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. % See the GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this macro package; if not, write to the % Free Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA. 11 dict begin /FontInfo 9 dict dup begin /version (001.104) readonly def /Notice (Copyright (C) 1996, 1997 Ross Moore and Y&Y, Inc.) readonly def /FullName (XYATIP10) readonly def /FamilyName (XYATIP) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -276 def /UnderlineThickness 138 def end readonly def /FontName /XYATIP10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /d0 put dup 15 /d15 put dup 30 /d30 put dup 34 /d34 put dup 47 /d47 put dup 56 /d56 put dup 60 /d60 put dup 63 /d63 put dup 64 /d64 put dup 79 /d79 put dup 94 /d94 put dup 111 /d111 put dup 127 /d127 put readonly def /FontBBox{-542 -542 542 542}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F409B932A74D078F654BDC704DE35B7CC455A4DC3 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b(a)f(sk)m(ew-symmetric)g(in)m(teger)g Fu(n)28 b Fr(\002)f Fu(n)40 b Fw(matrix)f Fu(B)45 b Fw(=)40 b(\()p Fu(b)3543 2524 y Fq(i;j)3623 2509 y Fw(\))94 2625 y(corresp)s(onds)48 b(to)e(a)g(quiv)m(er)h Fu(Q)p Fw(\()p Fu(B)5 b Fw(\))46 b(with)g(v)m(ertices)h(1)p Fu(;)17 b(:)g(:)g(:)f(;)h(n)p Fw(,)49 b(and)e Fu(b)2707 2640 y Fq(i;j)2834 2625 y Fw(arro)m(ws)f(from)f Fu(j)52 b Fw(to)46 b Fu(i)94 2742 y Fw(whenev)m(er)36 b Fu(b)564 2757 y Fq(i;j)673 2742 y Fu(>)29 b Fw(0.)45 b(F)-8 b(or)32 b(ev)m(ery)j(v)m(ertex)g Fu(k)s Fw(,)e(the)h Fv(mutation)40 b Fw(at)33 b Fu(k)j Fw(transforms)d Fu(B)38 b Fw(in)m(to)32 b(another)94 2867 y(sk)m(ew-symmetric)42 b(in)m(teger)e Fu(n)28 b Fr(\002)h Fu(n)40 b Fw(matrix)g Fu(\026)1816 2882 y Fq(k)1858 2867 y Fw(\()p Fu(B)5 b Fw(\))41 b(=)p 2172 2787 80 4 v 42 w Fu(B)46 b Fw(=)c(\()p 2448 2786 42 4 v Fu(b)2489 2882 y Fq(i;j)2569 2867 y Fw(\).)68 b(The)42 b(form)m(ula)c(for)p 3434 2786 V 41 w Fu(b)3475 2882 y Fq(i;j)3596 2867 y Fw(is)94 2983 y(giv)m(en)i(b)s(elo)m(w)g(in)f (\()799 2983 y SDict begin H.S end 799 2983 a Fw(7.4)923 2919 y SDict begin H.R end 923 2919 a 923 2983 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.7.4) cvn H.B /ANN pdfmark end 923 2983 a Fw(\).)65 b(It)40 b(is)f(w)m(ell-kno)m(wn)h (\(see)g(Prop)s(osition)2519 2983 y SDict begin H.S end 2519 2983 a Fw(7.1)2644 2919 y SDict begin H.R end 2644 2919 a 2644 2983 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.7.1) cvn H.B /ANN pdfmark end 2644 2983 a 40 w Fw(b)s(elo)m(w\))f (that)g(the)i(quiv)m(er)94 3104 y Fu(Q)p Fw(\()p 209 3024 80 4 v Fu(B)6 b Fw(\))32 b(can)h(b)s(e)g(obtained)f(from)f Fu(Q)p Fw(\()p Fu(B)5 b Fw(\))33 b(b)m(y)h(the)f(follo)m(wing)c (three-step)34 b(pro)s(cedure:)24 3291 y FA(Step)j(1.)42 b Fw(F)-8 b(or)40 b(ev)m(ery)j(incoming)c(arro)m(w)i Fu(a)i Fw(:)f Fu(j)48 b Fr(!)42 b Fu(k)i Fw(and)d(ev)m(ery)i(outgoing)d (arro)m(w)h Fu(b)i Fw(:)f Fu(k)j Fr(!)d Fu(i)p Fw(,)409 3407 y(create)i(a)f(\\comp)s(osite")e(arro)m(w)j([)p Fu(ba)p Fw(])i(:)g Fu(j)51 b Fr(!)46 b Fu(i)p Fw(;)i(th)m(us,)f(whenev) m(er)f Fu(b)2995 3422 y Fq(i;k)3082 3407 y Fu(;)17 b(b)3167 3422 y Fq(k)r(;j)3307 3407 y Fu(>)46 b Fw(0,)f(w)m(e)409 3524 y(create)33 b Fu(b)737 3539 y Fq(i;k)824 3524 y Fu(b)865 3539 y Fq(k)r(;j)993 3524 y Fw(new)g(arro)m(ws)g(from)f Fu(j)38 b Fw(to)32 b Fu(i)p Fw(.)24 3640 y FA(Step)37 b(2.)42 b Fw(Rev)m(erse)31 b(all)26 b(arro)m(ws)j(at)f Fu(k)s Fw(;)i(that)e(is,)h(replace)g(eac)m(h)g(arro)m(w)g Fu(a)f Fw(:)g Fu(j)33 b Fr(!)28 b Fu(k)j Fw(with)d Fu(a)3257 3604 y Fq(?)3325 3640 y Fw(:)f Fu(k)k Fr(!)c Fu(j)6 b Fw(,)409 3756 y(and)33 b Fu(b)28 b Fw(:)g Fu(k)i Fr(!)e Fu(i)k Fw(with)h Fu(b)1261 3720 y Fq(?)1328 3756 y Fw(:)28 b Fu(i)g Fr(!)f Fu(k)s Fw(.)24 3872 y FA(Step)37 b(3.)42 b Fw(If)30 b(some)g(orien)m(ted)h(2-cycles)f(app)s(ear)h(as)f(a)g (result)g(of)g(creating)g(new)h(arro)m(ws)g(in)f(Step)h(1,)409 3988 y(remo)m(v)m(e)d(an)m(y)g(of)f(them,)h(and)g(k)m(eep)h(remo)m (ving)d(these)j(2-cycles)f(one)f(b)m(y)i(one,)g(un)m(til)d(there)409 4105 y(are)33 b(no)f(more)g(left.)194 4291 y(In)37 b(the)h(case)f (where)h Fu(k)i Fw(is)d(a)f(source)i(or)f(a)f(sink)h(of)g Fu(Q)p Fw(\()p Fu(B)5 b Fw(\),)38 b(the)f(\014rst)g(and)g(last)f(steps) j(of)d(the)94 4413 y(ab)s(o)m(v)m(e)28 b(pro)s(cedure)g(are)f(not)g (applicable,)f(so)h Fu(Q)p Fw(\()p 1849 4333 V Fu(B)5 b Fw(\))27 b(is)f(obtained)h(from)e Fu(Q)p Fw(\()p Fu(B)5 b Fw(\))27 b(b)m(y)h(just)g(rev)m(ersing)94 4529 y(all)j(the)h(arro)m (ws)h(at)f Fu(k)s Fw(.)43 b(In)32 b(this)g(situation,)f(J.Bernstein,)i (I.)f(Gelfand,)g(and)g(V.)g(P)m(onomarev)g([)3585 4529 y SDict begin H.S end 3585 4529 a Fw(3)3634 4465 y SDict begin H.R end 3634 4465 a 3634 4529 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.bgp) cvn H.B /ANN pdfmark end 3634 4529 a Fw(])94 4645 y(in)m(tro)s(duced)24 b(the)g Fv(r)-5 b(e\015e)g(ction)26 b(functor)34 b Fw(at)24 b Fu(k)i Fw(sending)e(represen)m(tations)g(of)f(a)g(quiv)m(er)i Fu(Q)p Fw(\()p Fu(B)5 b Fw(\))23 b(\(with-)94 4766 y(out)33 b(relations\))f(in)m(to)g(represen)m(tations)j(of)d Fu(Q)p Fw(\()p 1804 4686 80 4 v Fu(B)6 b Fw(\).)45 b(A)33 b(mo)s(di\014cation) e(of)h(these)j(functors)e(acting)94 4883 y(on)f Fv(de)-5 b(c)g(or)g(ate)g(d)33 b(r)-5 b(epr)g(esentations)39 b Fw(w)m(as)32 b(in)m(tro)s(duced)g(in)f([)2143 4883 y SDict begin H.S end 2143 4883 a Fw(28)2241 4818 y SDict begin H.R end 2241 4818 a 2241 4883 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.mrz) cvn H.B /ANN pdfmark end 2241 4883 a Fw(])g(to)g(establish)h(a)f(link)f(b)s(et)m(w)m(een)k (clus-)94 4999 y(ter)j(algebras)f(and)h(quiv)m(er)g(represen)m(tations) h(\(the)f(de\014nition)f(of)g(decorated)h(represen)m(tations)94 5115 y(for)c(general)f(QPs)h(is)f(giv)m(en)h(b)s(elo)m(w)f(in)g (Section)1877 5115 y SDict begin H.S end 1877 5115 a Fw(10)1975 5051 y SDict begin H.R end 1975 5051 a 1975 5115 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (section.10) cvn H.B /ANN pdfmark end 1975 5115 a Fw(\).)194 5231 y(An)25 b(elemen)m(tary)g(approac)m(h)h(of)e([)1380 5231 y SDict begin H.S end 1380 5231 a Fw(28)1478 5167 y SDict begin H.R end 1478 5167 a 1478 5231 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.mrz) cvn H.B /ANN pdfmark end 1478 5231 a Fw(])h(has)g(not)g(b)s(een)h(further)f(pursued)i(un)m(til)d(no)m (w,)j(giving)c(w)m(a)m(y)94 5347 y(to)41 b(a)f(more)g(sophisticated)g (approac)m(h)h(via)f(cluster)g(categories)h(and)f(cluster-tilted)f (algebras)94 5464 y(dev)m(elop)s(ed)48 b(in)e([)716 5464 y SDict begin H.S end 716 5464 a Fw(7)765 5400 y SDict begin H.R end 765 5400 a 765 5464 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.BMRRT) cvn H.B /ANN pdfmark end 765 5464 a Fw(,)839 5464 y SDict begin H.S end 839 5464 a Fw(8)888 5400 y SDict begin H.R end 888 5400 a 888 5464 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.BMR1) cvn H.B /ANN pdfmark end 888 5464 a Fw(,)961 5464 y SDict begin H.S end 961 5464 a Fw(9)1010 5400 y SDict begin H.R end 1010 5400 a 1010 5464 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.BMR2) cvn H.B /ANN pdfmark end 1010 5464 a Fw(,)1084 5464 y SDict begin H.S end 1084 5464 a Fw(10)1182 5400 y SDict begin H.R end 1182 5400 a 1182 5464 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.BMR3) cvn H.B /ANN pdfmark end 1182 5464 a Fw(,)1255 5464 y SDict begin H.S end 1255 5464 a Fw(12)1353 5400 y SDict begin H.R end 1353 5400 a 1353 5464 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.CC) cvn H.B /ANN pdfmark end 1353 5464 a Fw(,)1427 5464 y SDict begin H.S end 1427 5464 a Fw(13)1524 5400 y SDict begin H.R end 1524 5400 a 1524 5464 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.CCS) cvn H.B /ANN pdfmark end 1524 5464 a Fw(,)1598 5464 y SDict begin H.S end 1598 5464 a Fw(14)1696 5400 y SDict begin H.R end 1696 5400 a 1696 5464 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.CK1) cvn H.B /ANN pdfmark end 1696 5464 a Fw(,)1769 5464 y SDict begin H.S end 1769 5464 a Fw(15)1867 5400 y SDict begin H.R end 1867 5400 a 1867 5464 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.CK2) cvn H.B /ANN pdfmark end 1867 5464 a Fw(])h(and)g(man)m(y)f(other)h (publications.)85 b(Most)47 b(of)94 5580 y(the)29 b(results)f(in)f (these)i(pap)s(ers)f(are)g(for)g(the)g(quiv)m(ers)h(obtained)e(b)m(y)i (m)m(utations)e(from)f(hereditary)94 5696 y(algebras)c(\(i.e.,)j(quiv)m (ers)f(without)e(orien)m(ted)h(cycles)g(and)g(without)f(relations\).)39 b(In)23 b(this)g(pap)s(er)f(w)m(e)p eop end end %%Page: 3 3 TeXDict begin HPSdict begin 3 2 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.3) cvn H.B /DEST pdfmark end 94 218 a 1179 w Fs(QUIVERS)31 b(WITH)i(POTENTIALS)g(I)1140 b(3)94 417 y Fw(return)25 b(to)e(a)h(more)f(elemen)m(tary)h(p)s(oin)m(t)f(of)h (view)g(of)f([)1986 417 y SDict begin H.S end 1986 417 a Fw(28)2084 353 y SDict begin H.R end 2084 353 a 2084 417 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.mrz) cvn H.B /ANN pdfmark end 2084 417 a Fw(])h(and)g(prop)s(ose)g(an)g (alternativ)m(e)f(approac)m(h)94 533 y(\(whic)m(h)32 b(is)f(in)f(fact)h(more)f(general,)h(since)h(w)m(e)g(do)f(not)g(imp)s (ose)f(an)m(y)i(restrictions)f(on)g(quiv)m(ers)h(in)94 649 y(question\).)45 b(In)33 b(this)f(approac)m(h,)i(the)f(m)m (utations)f(at)g(arbitrary)g(v)m(ertices)i(\(not)e(just)h(sources)i(or) 94 766 y(sinks\))c(are)e(de\014ned)i(for)e(QPs)i(and)e(their)g (decorated)i(represen)m(tations.)44 b(The)30 b(construction)g(for)94 882 y(QPs)j(is)f(carried)g(out)g(in)f(Section)1357 882 y SDict begin H.S end 1357 882 a Fw(5)1406 818 y SDict begin H.R end 1406 818 a 1406 882 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (section.5) cvn H.B /ANN pdfmark end 1406 882 a Fw(,)h(and)h(for)e(their)h(represen)m(tations)h(in)f(Section)3166 882 y SDict begin H.S end 3166 882 a Fw(10)3263 818 y SDict begin H.R end 3263 818 a 3263 882 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (section.10) cvn H.B /ANN pdfmark end 3263 882 a Fw(.)43 b(It)33 b(turns)94 998 y(out)k(to)g(b)s(e)g (rather)g(delicate)f(and)h(requires)h(a)f(lot)e(of)i(tec)m(hnical)f (preparation.)56 b(The)38 b(\014rst)g(t)m(w)m(o)94 1114 y(steps)32 b(of)e(the)g(ab)s(o)m(v)m(e)h(m)m(utation)e(pro)s(cedure)i (extend)g(to)f(QPs)h(in)e(a)h(relativ)m(ely)f(straigh)m(tforw)m(ard)94 1231 y(w)m(a)m(y)-8 b(,)47 b(but)c(Step)h(3)e(presen)m(ts)j(a)e(real)f (c)m(hallenge:)63 b(w)m(e)44 b(need)g(to)f(accompan)m(y)g(the)g(remo)m (v)-5 b(al)42 b(of)94 1347 y(orien)m(ted)31 b(2-cycles)g(from)f(a)g (quiv)m(er)i(with)e(a)h(suitable)e(mo)s(di\014cation)f(of)j(the)g(p)s (oten)m(tial,)e(lea)m(ving)94 1463 y(the)44 b(corresp)s(onding)e (Jacobian)g(algebra)g(unc)m(hanged.)75 b(Our)42 b(main)f(device)j(in)e (dealing)f(with)94 1579 y(this)47 b(di\016cult)m(y)g(is)f(Theorem)1258 1579 y SDict begin H.S end 1258 1579 a Fw(4.6)1383 1515 y SDict begin H.R end 1383 1515 a 1383 1579 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.6) cvn H.B /ANN pdfmark end 1383 1579 a Fw(,)k(whic)m(h)d(is)g(the)g(crucial)e(tec)m(hnical) i(result)g(of)f(the)h(pap)s(er.)94 1696 y(Roughly)38 b(sp)s(eaking,)h(Theorem)1342 1696 y SDict begin H.S end 1342 1696 a Fw(4.6)1466 1631 y SDict begin H.R end 1466 1631 a 1466 1696 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.6) cvn H.B /ANN pdfmark end 1466 1696 a 38 w Fw(asserts)h(that)d(ev) m(ery)j(p)s(oten)m(tial)c Fu(S)44 b Fw(can)39 b(b)s(e)f(transformed)94 1812 y(b)m(y)29 b(an)e(automorphism)f(of)g(the)i(path)g(algebra)e(in)m (to)h(the)h(sum)f(of)g(t)m(w)m(o)h(p)s(oten)m(tials)e Fu(S)3170 1827 y Fp(triv)3315 1812 y Fw(and)h Fu(S)3559 1827 y Fp(red)94 1928 y Fw(on)35 b(the)f(disjoin)m(t)f(sets)i(of)f (arro)m(ws,)h(where)h(the)e Fv(trivial)45 b Fw(part)34 b Fu(S)2420 1943 y Fp(triv)2571 1928 y Fw(is)f(a)h(linear)f(com)m (bination)f(of)94 2044 y(cyclic)37 b(2-paths,)h(while)e(the)h Fv(r)-5 b(e)g(duc)g(e)g(d)46 b Fw(part)37 b Fu(S)1808 2059 y Fp(red)1947 2044 y Fw(in)m(v)m(olv)m(es)g(only)g(cyclic)f Fu(d)p Fw(-paths)g(with)h Fu(d)d Fr(\025)i Fw(3.)94 2160 y(F)-8 b(urthermore,)33 b(the)g(Jacobian)e(algebra)h(of)g Fu(S)1775 2175 y Fp(red)1910 2160 y Fw(is)g(isomorphic)e(to)j(that)f (of)g Fu(S)6 b Fw(.)194 2277 y(Sev)m(eral)42 b(commen)m(ts)g(on)g(this) f(result)h(are)f(in)g(order.)71 b(First,)43 b(our)f(argumen)m(ts)g(hea) m(vily)f(de-)94 2393 y(p)s(end)29 b(on)f(the)g(setup)h(using)e (completed)h(path)g(algebras,)g(th)m(us)h(allo)m(wing)c(p)s(oten)m (tials)i(to)g(in)m(v)m(olv)m(e)94 2509 y(in\014nite)44 b(sums)h(of)f(cyclic)g(paths.)80 b(Second,)49 b(the)c(reduction)f Fu(S)54 b Fr(7!)47 b Fu(S)2756 2524 y Fp(red)2903 2509 y Fw(is)d(not)h(giv)m(en)f(b)m(y)i(a)94 2625 y(canonical)33 b(pro)s(cedure.)50 b(As)35 b(a)f(consequence,)k(our)d(construction)f (of)g(m)m(utations)f(for)h(QPs)h(and)94 2742 y(their)c(represen)m (tations)h(is)f(not)f(functorial)f(in)i(an)m(y)g(ob)m(vious)g(sense.)45 b(On)31 b(the)h(p)s(ositiv)m(e)e(side,)h(w)m(e)94 2858 y(pro)m(v)m(e)36 b(that)e(ev)m(ery)i(m)m(utation)d(is)g(a)h(w)m (ell-de\014ned)h(transformation)d(on)i(the)g(righ)m(t-equiv)-5 b(alence)94 2974 y(classes)43 b(of)f(QPs)h(\(and)f(their)f(represen)m (tations\),)46 b(where,)f(roughly)d(sp)s(eaking,)i(t)m(w)m(o)f(QPs)f (are)94 3090 y Fv(right-e)-5 b(quivalent)41 b Fw(if)30 b(they)i(can)g(b)s(e)f(obtained)g(from)f(eac)m(h)j(other)e(b)m(y)h(an)g (automorphism)d(of)i(the)94 3207 y(path)i(algebra)f(\(for)f(more)h (precise)i(de\014nitions)e(see)h(De\014nitions)2537 3207 y SDict begin H.S end 2537 3207 a Fw(4.2)2661 3142 y SDict begin H.R end 2661 3142 a 2661 3207 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.2) cvn H.B /ANN pdfmark end 2661 3207 a 33 w Fw(and)2884 3207 y SDict begin H.S end 2884 3207 a Fw(10.2)3057 3142 y SDict begin H.R end 3057 3142 a 3057 3207 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.10.2) cvn H.B /ANN pdfmark end 3057 3207 a Fw(\).)194 3323 y(Finally)-8 b(,)32 b(it)g(is)i(imp)s(ortan)m(t)d(to)j(k)m(eep)h (in)e(mind)g(that,)h(ev)m(en)h(with)f(the)g(help)f(of)h(Theorem)3510 3323 y SDict begin H.S end 3510 3323 a Fw(4.6)3634 3259 y SDict begin H.R end 3634 3259 a 3634 3323 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.6) cvn H.B /ANN pdfmark end 3634 3323 a Fw(,)94 3439 y(in)i(order)g(to)g(get)g(rid)g(of)f (all)f(orien)m(ted)j(2-cycles)f(in)g(the)g(m)m(utated)g(QP)-8 b(,)37 b(one)f(needs)i(to)e(imp)s(ose)94 3555 y(some)29 b(\\genericit)m(y")f(conditions)g(on)g(the)h(initial)c(p)s(oten)m(tial) i Fu(S)6 b Fw(.)42 b(These)31 b(conditions)c(are)i(studied)94 3671 y(in)35 b(Section)556 3671 y SDict begin H.S end 556 3671 a Fw(7)605 3607 y SDict begin H.R end 605 3607 a 605 3671 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (section.7) cvn H.B /ANN pdfmark end 605 3671 a Fw(.)52 b(They)37 b(are)f(not)f(v)m(ery)j(explicit)c(in)h(general,)h(but)f(w)m(e)i(in)m (tro)s(duce)f(an)f(imp)s(ortan)m(t)94 3788 y(class)k(or)f Fv(rigid)48 b Fw(QPs)39 b(\(see)g(De\014nitions)1616 3788 y SDict begin H.S end 1616 3788 a Fw(6.7)1741 3723 y SDict begin H.R end 1741 3723 a 1741 3788 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.6.7) cvn H.B /ANN pdfmark end 1741 3788 a 38 w Fw(and)1974 3788 y SDict begin H.S end 1974 3788 a Fw(6.10)2147 3723 y SDict begin H.R end 2147 3723 a 2147 3788 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.6.10) cvn H.B /ANN pdfmark end 2147 3788 a Fw(\))g(for)e(whic)m(h)i(the)g(absence)h(of)d(orien)m(ted)94 3904 y(2-cycles)c(after)g(a)f(QP)h(m)m(utation)e(is)h(guaran)m(teed.) 194 4020 y(W)-8 b(e)23 b(no)m(w)g(describ)s(e)h(the)f(con)m(ten)m(ts)h (of)e(the)h(pap)s(er)g(in)f(more)f(detail.)39 b(In)23 b(Section)3055 4020 y SDict begin H.S end 3055 4020 a Fw(2)3104 3956 y SDict begin H.R end 3104 3956 a 3104 4020 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (section.2) cvn H.B /ANN pdfmark end 3104 4020 a 23 w Fw(w)m(e)g(in)m(tro)s(duce)94 4136 y(an)h(algebraic)d(setup)j(for)f(dealing)f(with)h(quiv)m(ers)h (and)g(their)e(path)h(algebras.)40 b(W)-8 b(e)24 b(\014x)g(a)f(base)h (\014eld)94 4253 y Fu(K)7 b Fw(,)27 b(and)e(enco)s(de)g(a)g(quiv)m(er)g (with)f(the)i(v)m(ertex)g(set)f Fu(Q)1978 4268 y Fp(0)2043 4253 y Fw(and)f(the)i(arro)m(w)e(set)i Fu(Q)2872 4268 y Fp(1)2936 4253 y Fw(b)m(y)g(its)e Fv(vertex)j(sp)-5 b(an)94 4369 y Fu(R)37 b Fw(=)f Fu(K)407 4333 y Fq(Q)463 4342 y Fo(0)539 4369 y Fw(and)h Fv(arr)-5 b(ow)39 b(sp)-5 b(an)44 b Fu(A)36 b Fw(=)g Fu(K)1559 4333 y Fq(Q)1615 4342 y Fo(1)1654 4369 y Fw(.)57 b(Th)m(us,)41 b Fu(R)d Fw(is)f(a)g(\014nite-dimensional)d(comm)m(utativ)m(e)94 4485 y Fu(K)7 b Fw(-algebra,)39 b(and)f Fu(A)g Fw(is)g(a)g (\014nite-dimensional)d Fu(R)q Fw(-bimo)s(dule.)58 b(W)-8 b(e)39 b(then)g(in)m(tro)s(duce)f(the)h Fv(p)-5 b(ath)94 4601 y(algebr)g(a)1546 4910 y Fu(R)q Fr(h)p Fu(A)p Fr(i)27 b Fw(=)1942 4785 y Fn(1)1902 4815 y Fm(M)1915 5027 y Fq(d)p Fp(=0)2070 4910 y Fu(A)2143 4869 y Fq(d)2183 4910 y Fu(;)94 5253 y Fw(and,)33 b(more)f(imp)s(ortan)m(tly)e(for)i(our)h (purp)s(oses,)h(the)f Fv(c)-5 b(omplete)34 b(p)-5 b(ath)34 b(algebr)-5 b(a)1518 5602 y Fu(R)q Fr(hh)p Fu(A)p Fr(ii)27 b Fw(=)1981 5477 y Fn(1)1953 5507 y Fm(Y)1953 5719 y Fq(d)p Fp(=0)2097 5602 y Fu(A)2170 5561 y Fq(d)2210 5602 y Fw(;)p eop end end %%Page: 4 4 TeXDict begin HPSdict begin 4 3 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.4) cvn H.B /DEST pdfmark end 94 218 a Fs(4)530 b(HARM)25 b(DERKSEN,)f(JERZY)i(WEYMAN,)f(AND)f(ANDREI)g(ZELEVINSKY)94 417 y Fw(here)34 b Fu(A)379 381 y Fq(d)451 417 y Fw(stands)g(for)d(the) i Fu(d)p Fw(-fold)d(tensor)j(p)s(o)m(w)m(er)g(of)f Fu(A)h Fw(as)f(an)g Fu(R)q Fw(-bimo)s(dule.)41 b(W)-8 b(e)33 b(view)f Fu(R)q Fr(hh)p Fu(A)p Fr(ii)94 533 y Fw(as)38 b(a)g(top)s(ological)c Fu(R)q Fw(-algebra)i(via)h(the)h Fl(m)p Fw(-adic)e(top)s(ology)-8 b(,)38 b(where)h Fl(m)f Fw(is)f(the)h(t)m(w)m(o-sided)g(ideal)94 649 y(generated)c(b)m(y)f Fu(A)p Fw(.)194 766 y(In)48 b(Section)688 766 y SDict begin H.S end 688 766 a Fw(3)737 701 y SDict begin H.R end 737 701 a 737 766 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (section.3) cvn H.B /ANN pdfmark end 737 766 a 47 w Fw(w)m(e)h(in)m (tro)s(duce)f(some)g(of)f(our)h(main)e(ob)5 b(jects)49 b(of)e(study:)76 b(p)s(oten)m(tials)46 b(and)94 882 y(their)34 b(Jacobian)f(ideals.)47 b(It)34 b(is)f(natural)g(to)h(view)g(p)s(oten)m (tials)f(as)h(elemen)m(ts)g(of)g(the)g Fv(tr)-5 b(ac)g(e)36 b(sp)-5 b(ac)g(e)94 998 y Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fu(=)p Fr(f)p Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fu(;)17 b(R)q Fr(hh)p Fu(A)p Fr(iig)p Fw(,)30 b(where)j Fr(f)p Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fu(;)17 b(R)q Fr(hh)p Fu(A)p Fr(iig)30 b Fw(is)i(the)h(closure)g(of)f(the)h(v)m(ector)g(sub-)94 1114 y(space)i(in)d Fu(R)q Fr(hh)p Fu(A)p Fr(ii)g Fw(spanned)i(b)m(y)g (all)d(comm)m(utators.)44 b(It)33 b(is)f(more)h(con)m(v)m(enien)m(t)h (for)f(us)g(to)g(de\014ne)94 1231 y(a)j(p)s(oten)m(tial)d Fu(S)41 b Fw(as)36 b(an)f(elemen)m(t)g(of)g(the)h(cyclic)e(part)i(of)e Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fw(;)i(for)f(all)e(practical)h(purp)s (oses,)94 1347 y Fu(S)h Fw(can)30 b(b)s(e)g(replaced)f(b)m(y)h(a)f Fv(cyclic)-5 b(al)5 b(ly)32 b(e)-5 b(quivalent)38 b Fw(p)s(oten)m (tial,)28 b(that)h(is,)h(the)f(one)h(with)f(the)h(same)94 1463 y(image)42 b(in)g(the)h(trace)g(space.)76 b(T)-8 b(o)42 b(de\014ne)i(the)g(Jacobian)e(ideal)f Fu(J)9 b Fw(\()p Fu(S)d Fw(\))43 b(and)g(deriv)m(e)g(its)g(basic)94 1579 y(prop)s(erties,)31 b(w)m(e)h(dev)m(elop)f(the)g(formalism)c(of)j Fv(cyclic)i(derivatives)p Fw(,)e(in)g(particular,)f(establishing)94 1696 y(\\cyclic")g(v)m(ersions)h(of)e(the)i(Leibniz)e(rule)h(and)g(the) g(c)m(hain)g(rule.)42 b(The)30 b(main)d(result)i(of)g(Section)3613 1696 y SDict begin H.S end 3613 1696 a Fw(3)3661 1631 y SDict begin H.R end 3661 1631 a 3661 1696 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (section.3) cvn H.B /ANN pdfmark end 3661 1696 a 94 1812 a Fw(is)34 b(Prop)s(osition)719 1812 y SDict begin H.S end 719 1812 a Fw(3.7)844 1748 y SDict begin H.R end 844 1748 a 844 1812 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.3.7) cvn H.B /ANN pdfmark end 844 1812 a 34 w Fw(that)f(asserts)i(that)e(an)m(y)h(isomorphism)d Fu(')j Fw(of)f(path)h(algebras)e(sends)k Fu(J)9 b Fw(\()p Fu(S)d Fw(\))94 1928 y(to)33 b Fu(J)9 b Fw(\()p Fu(')p Fw(\()p Fu(S)d Fw(\)\).)43 b(Note)33 b(that)g(cyclic)f(deriv)-5 b(ativ)m(es)33 b(for)f(general)g(non-comm)m(utativ)m(e)g(algebras)g(w)m (ere)94 2044 y(in)m(tro)s(duced)39 b(in)e([)734 2044 y SDict begin H.S end 734 2044 a Fw(29)831 1980 y SDict begin H.R end 831 1980 a 831 2044 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.rss) cvn H.B /ANN pdfmark end 831 2044 a Fw(],)j(and)e(the)g(results)h(w)m(e)g(presen)m(t)g(can)g(b)s(e)f (easily)f(deduced)j(from)c(those)j(in)94 2160 y(lo)s(c.cit.)48 b(F)-8 b(or)33 b(the)i(con)m(v)m(enience)h(of)e(the)h(reader,)g(w)m(e)h (presen)m(t)g(complete)d(indep)s(enden)m(t)j(pro)s(ofs.)94 2277 y(Victor)f(Ginzburg)g(informed)e(us)j(that)g(in)e(the)i(con)m (text)h(of)d(path)i(algebras)e(of)h(quiv)m(ers,)j(cyclic)94 2393 y(deriv)-5 b(ativ)m(es)46 b(w)m(ere)h(in)m(tro)s(duced)f(and)g (studied)g(in)f([)2044 2393 y SDict begin H.S end 2044 2393 a Fw(5)2093 2329 y SDict begin H.R end 2093 2329 a 2093 2393 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.blb) cvn H.B /ANN pdfmark end 2093 2393 a Fw(,)2165 2393 y SDict begin H.S end 2165 2393 a Fw(24)2263 2329 y SDict begin H.R end 2263 2329 a 2263 2393 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.gi1) cvn H.B /ANN pdfmark end 2263 2393 a Fw(],)k(and)d(that)f(Prop)s(osition)3331 2393 y SDict begin H.S end 3331 2393 a Fw(3.7)3456 2329 y SDict begin H.R end 3456 2329 a 3456 2393 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.3.7) cvn H.B /ANN pdfmark end 3456 2393 a 45 w Fw(is)h(a)94 2509 y(consequence)e(of)c(the)h (geometric)e(in)m(terpretation)h(of)g Fu(J)9 b Fw(\()p Fu(S)d Fw(\))40 b(giv)m(en)h(in)f([)2805 2509 y SDict begin H.S end 2805 2509 a Fw(25)2902 2445 y SDict begin H.R end 2902 2445 a 2902 2509 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.gi) cvn H.B /ANN pdfmark end 2902 2509 a Fw(,)j(De\014nition)c(5.1.1,)94 2625 y(Lemma)32 b(5.1.3].)194 2742 y(In)48 b(Section)687 2742 y SDict begin H.S end 687 2742 a Fw(4)735 2677 y SDict begin H.R end 735 2677 a 735 2742 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (section.4) cvn H.B /ANN pdfmark end 735 2742 a 47 w Fw(w)m(e)h(in)m(tro)s(duce)e(quiv)m(ers)h(with)f(p)s(oten)m(tials)f (\(QPs\))h(and)h(de\014ne)g(the)g(righ)m(t-)94 2858 y(equiv)-5 b(alence)34 b(relation)e(on)i(them,)g(whic)m(h)g(pla)m(ys)g(an)f(imp)s (ortan)m(t)f(role)h(in)g(the)h(pap)s(er.)47 b(W)-8 b(e)34 b(then)94 2974 y(state)c(and)e(pro)m(v)m(e)i(the)f(k)m(ey)h(tec)m (hnical)e(result)h(of)f(the)h(pap)s(er:)42 b(Splitting)26 b(Theorem)3169 2974 y SDict begin H.S end 3169 2974 a Fw(4.6)3293 2910 y SDict begin H.R end 3293 2910 a 3293 2974 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.6) cvn H.B /ANN pdfmark end 3293 2974 a Fw(,)k(already)94 3090 y(discussed)k(ab)s(o)m(v)m(e.)44 b(The)32 b(pro)s(of)e(is)h (elemen)m(tary)h(but)f(prett)m(y)i(in)m(v)m(olv)m(ed;)f(it)f(uses)h(in) f(an)g(essen)m(tial)94 3207 y(w)m(a)m(y)c(the)f(top)s(ology)e(in)g(a)h (complete)g(path)g(algebra.)40 b(In)26 b(order)f(not)g(to)g(in)m (terrupt)h(the)g(argumen)m(t,)94 3323 y(w)m(e)32 b(mo)m(v)m(e)e(to)g (the)g(App)s(endix)h(our)f(treatmen)m(t)g(of)g(the)g(top)s(ological)d (prop)s(erties)j(needed)h(for)f(the)94 3439 y(pro)s(of)i(of)g(one)h(of) f(the)h(tec)m(hnical)f(lemmas.)194 3555 y(In)38 b(Section)668 3555 y SDict begin H.S end 668 3555 a Fw(5)717 3491 y SDict begin H.R end 717 3491 a 717 3555 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (section.5) cvn H.B /ANN pdfmark end 717 3555 a 37 w Fw(w)m(e)h(\014nally)e(in)m(tro)s(duce)h(the)g(m)m (utations)f(of)g(QPs.)60 b(Using)37 b(Theorem)3360 3555 y SDict begin H.S end 3360 3555 a Fw(4.6)3484 3491 y SDict begin H.R end 3484 3491 a 3484 3555 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.6) cvn H.B /ANN pdfmark end 3484 3555 a Fw(,)i(w)m(e)94 3671 y(pro)m(v)m(e)30 b(that)f(the)g(m)m (utation)e(at)i(an)f(arbitrary)g(v)m(ertex)i(is)f(a)f(w)m (ell-de\014ned)h(in)m(v)m(olution)e(on)i(the)g(set)94 3788 y(of)k(righ)m(t-equiv)-5 b(alence)31 b(classes)j(of)e(reduced)i (QPs)f(\(Theorem)2404 3788 y SDict begin H.S end 2404 3788 a Fw(5.7)2529 3723 y SDict begin H.R end 2529 3723 a 2529 3788 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.5.7) cvn H.B /ANN pdfmark end 2529 3788 a Fw(\).)194 3904 y(In)k(Section)665 3904 y SDict begin H.S end 665 3904 a Fw(6)714 3840 y SDict begin H.R end 714 3840 a 714 3904 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (section.6) cvn H.B /ANN pdfmark end 714 3904 a Fw(,)g(w)m(e)g(study)h(some)e(m)m (utation)e(in)m(v)-5 b(arian)m(ts)36 b(of)g(QPs.)55 b(In)36 b(particular,)g(w)m(e)h(sho)m(w)94 4020 y(that)f(m)m(utations)f (preserv)m(e)j(the)e(class)g(of)f(QPs)i(with)e(\014nite-dimensional)e (Jacobian)i(algebras)94 4136 y(\(Corollary)558 4136 y SDict begin H.S end 558 4136 a Fw(6.6)682 4072 y SDict begin H.R end 682 4072 a 682 4136 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.6.6) cvn H.B /ANN pdfmark end 682 4136 a Fw(\).)42 b(Another)28 b(imp)s(ortan)m(t)d(prop)s(ert)m(y)j(of)e (QPs)i(preserv)m(ed)h(b)m(y)f(m)m(utations)e(is)h Fv(rigid-)94 4253 y(ity)43 b Fw(\(Corollary)716 4253 y SDict begin H.S end 716 4253 a Fw(6.11)889 4188 y SDict begin H.R end 889 4188 a 889 4253 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.6.11) cvn H.B /ANN pdfmark end 889 4253 a Fw(\),)34 b(whic)m(h)g(w)m(as)h(already)e(men)m(tioned)g(ab)s(o)m(v)m(e.)47 b(F)-8 b(or)33 b(the)h(precise)g(de\014nition)94 4369 y(of)d(rigid)f(QPs)i(see)g(De\014nitions)1290 4369 y SDict begin H.S end 1290 4369 a Fw(6.7)1415 4305 y SDict begin H.R end 1415 4305 a 1415 4369 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.6.7) cvn H.B /ANN pdfmark end 1415 4369 a 31 w Fw(and)1635 4369 y SDict begin H.S end 1635 4369 a Fw(6.10)1808 4305 y SDict begin H.R end 1808 4305 a 1808 4369 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.6.10) cvn H.B /ANN pdfmark end 1808 4369 a 31 w Fw(b)s(elo)m(w;)g(in)m (tuitiv)m(ely)-8 b(,)30 b(a)h(QP)h(is)e(rigid)g(if)g(its)h(righ)m(t-)94 4485 y(equiv)-5 b(alence)33 b(class)g(is)f(in)m(v)-5 b(arian)m(t)31 b(under)j(in\014nitesimal)29 b(deformations.)194 4601 y(In)35 b(Section)660 4601 y SDict begin H.S end 660 4601 a Fw(7)709 4537 y SDict begin H.R end 709 4537 a 709 4601 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (section.7) cvn H.B /ANN pdfmark end 709 4601 a Fw(,)g(w)m(e)g(in)m(tro)s(duce) f(and)g(study)h Fv(nonde)-5 b(gener)g(ate)40 b Fw(QPs,)35 b(that)f(is,)g(those)h(to)f(whic)m(h)94 4718 y(one)29 b(can)f(apply)f(an)h(arbitrary)f(sequence)j(of)e(m)m(utations)e (without)i(creating)f(orien)m(ted)h(2-cycles.)94 4834 y(In)49 b(Corollary)678 4834 y SDict begin H.S end 678 4834 a Fw(7.4)802 4770 y SDict begin H.R end 802 4770 a 802 4834 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.7.4) cvn H.B /ANN pdfmark end 802 4834 a 48 w Fw(w)m(e)g(sho)m(w)g(that) e(nondegeneracy)j(is)d(guaran)m(teed)i(b)m(y)g(non-v)-5 b(anishing)46 b(of)94 4950 y(coun)m(tably)41 b(man)m(y)g(nonzero)g(p)s (olynomial)c(functions)k(on)f(the)i(space)f(of)f(p)s(oten)m(tials.)67 b(In)41 b(par-)94 5066 y(ticular,)50 b(if)d(the)h(base)g(\014eld)f Fu(K)55 b Fw(is)47 b(uncoun)m(table,)52 b(a)47 b(nondegenerate)i(QP)f (exists)g(for)g(ev)m(ery)94 5182 y(underlying)33 b(quiv)m(er.)194 5299 y(Section)525 5299 y SDict begin H.S end 525 5299 a Fw(8)574 5234 y SDict begin H.R end 574 5234 a 574 5299 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (section.8) cvn H.B /ANN pdfmark end 574 5299 a 21 w Fw(con)m(tains)22 b(some)g(examples)g(of)f(rigid)f(and)i(non-rigid)e(p)s(oten)m(tials)g (and)i(some)g(further)94 5415 y(results)h(illustrating)d(the)j(imp)s (ortance)e(of)h(rigidit)m(y)-8 b(.)38 b(A)22 b(simple)f(but)i(imp)s (ortan)m(t)e(Prop)s(osition)3537 5415 y SDict begin H.S end 3537 5415 a Fw(8.1)3661 5351 y SDict begin H.R end 3661 5351 a 3661 5415 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.8.1) cvn H.B /ANN pdfmark end 3661 5415 a 94 5531 a Fw(asserts)35 b(that)f(rigid)d(QPs)k(ha)m(v)m(e)g(no)e(orien)m(ted)h (2-cycles.)47 b(Com)m(bining)32 b(this)h(with)g(the)h(fact)g(that)94 5647 y(rigidit)m(y)24 b(is)h(preserv)m(ed)j(b)m(y)f(m)m(utations,)f(w)m (e)h(conclude)f(that)g(ev)m(ery)h(rigid)d(QP)i(is)f(nondegenerate.)p eop end end %%Page: 5 5 TeXDict begin HPSdict begin 5 4 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.5) cvn H.B /DEST pdfmark end 94 218 a 1179 w Fs(QUIVERS)31 b(WITH)i(POTENTIALS)g(I)1140 b(5)94 417 y Fw(Using)34 b(a)g(result)f(b)m(y)i(Keller-Reiten)d([)1488 417 y SDict begin H.S end 1488 417 a Fw(27)1586 353 y SDict begin H.R end 1586 353 a 1586 417 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.kr) cvn H.B /ANN pdfmark end 1586 417 a Fw(],)i(w)m(e)h(sho)m(w)g(in)e(Example)2583 417 y SDict begin H.S end 2583 417 a Fw(8.7)2708 353 y SDict begin H.R end 2708 353 a 2708 417 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.8.7) cvn H.B /ANN pdfmark end 2708 417 a 34 w Fw(that)h(the)g(class)g(of)f(rigid)94 533 y(QPs)28 b(\(as)f(w)m(ell)f(as)h(the)g(class)g(of)f(QPs)h(with)f (\014nite-dimensional)e(Jacobian)i(algebras\))g(is)g(strictly)94 649 y(greater)e(than)g(the)g(class)f(of)h(QPs)g(m)m(utation-equiv)-5 b(alen)m(t)21 b(to)i(acyclic)g(ones.)42 b(On)23 b(the)h(other)g(hand,) 94 766 y(Example)504 766 y SDict begin H.S end 504 766 a Fw(8.6)629 701 y SDict begin H.R end 629 701 a 629 766 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.8.6) cvn H.B /ANN pdfmark end 629 766 a 37 w Fw(exhibits)37 b(an)g(underlying)f (quiv)m(er)i(without)f(orien)m(ted)g(2-cycles)g(that)g(do)s(es)h(not)94 882 y(admit)c(a)h(rigid)f(QP;)i(th)m(us,)h(the)f(class)f(of)g (nondegenerate)i(QPs)f(is)f(strictly)g(greater)g(than)h(the)94 998 y(class)d(of)f(rigid)f(ones.)194 1114 y(In)d(Section)648 1114 y SDict begin H.S end 648 1114 a Fw(9)697 1050 y SDict begin H.R end 697 1050 a 697 1114 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (section.9) cvn H.B /ANN pdfmark end 697 1114 a Fw(,)h(w)m(e)g(consider)f(quiv)m(ers)h(that)f(are)g(m)m (utation-equiv)-5 b(alen)m(t)25 b(to)j(a)g(Dynkin)f(quiv)m(er.)94 1231 y(F)-8 b(or)30 b(ev)m(ery)j(suc)m(h)f(underlying)f(quiv)m(er,)h(w) m(e)g(compute)e(explicitly)f(the)j(corresp)s(onding)e(rigid)f(QP)94 1347 y(\(Prop)s(osition)660 1347 y SDict begin H.S end 660 1347 a Fw(9.1)784 1283 y SDict begin H.R end 784 1283 a 784 1347 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.9.1) cvn H.B /ANN pdfmark end 784 1347 a Fw(\).)51 b(Comparing)34 b(this)g(result)h(with)g(the)g(description)g(of)f(cluster-tilted)g (alge-)94 1463 y(bras)k(obtained)g(in)f([)864 1463 y SDict begin H.S end 864 1463 a Fw(13)961 1399 y SDict begin H.R end 961 1399 a 961 1463 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.CCS) cvn H.B /ANN pdfmark end 961 1463 a Fw(,)1026 1463 y SDict begin H.S end 1026 1463 a Fw(9)1075 1399 y SDict begin H.R end 1075 1399 a 1075 1463 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.BMR2) cvn H.B /ANN pdfmark end 1075 1463 a Fw(],)i(w)m(e)g(conclude)f(in)f (Corollary)2278 1463 y SDict begin H.S end 2278 1463 a Fw(9.3)2402 1399 y SDict begin H.R end 2402 1399 a 2402 1463 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.9.3) cvn H.B /ANN pdfmark end 2402 1463 a 38 w Fw(that)g(in)g(the)h(case)h (in)e(question,)94 1579 y(ev)m(ery)k(cluster-tilted)c(algebra)h(can)h (b)s(e)g(iden)m(ti\014ed)g(with)f(the)i(Jacobian)e(algebra)f(of)i(the)g (cor-)94 1696 y(resp)s(onding)i(rigid)f(QP)-8 b(.)41 b(Th)m(us,)k(Jacobian)40 b(algebras)g(can)i(b)s(e)f(view)m(ed)h(as)g (generalizations)d(of)94 1812 y(cluster-tilted)31 b(algebras.)194 1928 y(In)f(Section)651 1928 y SDict begin H.S end 651 1928 a Fw(10)748 1864 y SDict begin H.R end 748 1864 a 748 1928 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (section.10) cvn H.B /ANN pdfmark end 748 1928 a 29 w Fw(w)m(e)g(in)m(tro)s(duce)g (decorated)f(represen)m(tations)i(of)e(QPs)g(\(QP-represen)m(tations,) 94 2044 y(for)h(short\))g(and)g(their)g(righ)m(t-equiv)-5 b(alence)29 b(\(De\014nitions)2219 2044 y SDict begin H.S end 2219 2044 a Fw(10.1)2393 1980 y SDict begin H.R end 2393 1980 a 2393 2044 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.10.1) cvn H.B /ANN pdfmark end 2393 2044 a 30 w Fw(and)2610 2044 y SDict begin H.S end 2610 2044 a Fw(10.2)2783 1980 y SDict begin H.R end 2783 1980 a 2783 2044 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.10.2) cvn H.B /ANN pdfmark end 2783 2044 a Fw(\).)43 b(W)-8 b(e)30 b(then)h(presen)m(t)h(a)94 2160 y(represen)m(tation-theoretic)i (extension)g(of)f(Splitting)e(Theorem)2498 2160 y SDict begin H.S end 2498 2160 a Fw(4.6)2623 2096 y SDict begin H.R end 2623 2096 a 2623 2160 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.6) cvn H.B /ANN pdfmark end 2623 2160 a 33 w Fw(b)m(y)k(de\014ning)e(the)h(reduced)94 2277 y(part)48 b(of)f(a)h(QP-represen)m(tation)g Fr(M)g Fw(\(De\014nition)2046 2277 y SDict begin H.S end 2046 2277 a Fw(10.4)2219 2212 y SDict begin H.R end 2219 2212 a 2219 2277 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.10.4) cvn H.B /ANN pdfmark end 2219 2277 a Fw(\))g(and)g(pro)m(ving)f (that,)52 b(up)c(to)g(righ)m(t-)94 2393 y(equiv)-5 b(alence,)31 b(it)e(is)h(determined)g(b)m(y)h(the)f(righ)m(t-equiv)-5 b(alence)29 b(class)i(of)e Fr(M)h Fw(\(Prop)s(osition)3423 2393 y SDict begin H.S end 3423 2393 a Fw(10.5)3596 2329 y SDict begin H.R end 3596 2329 a 3596 2393 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.10.5) cvn H.B /ANN pdfmark end 3596 2393 a Fw(\).)94 2509 y(W)-8 b(e)48 b(use)g(this)f(result)g (to)g(in)m(tro)s(duce)g(m)m(utations)f(of)h(QP-represen)m(tations)h (and)f(to)g(pro)m(v)m(e)i(a)94 2625 y(represen)m(tation-theoretic)38 b(extension)f(of)g(Theorem)2110 2625 y SDict begin H.S end 2110 2625 a Fw(5.7)2235 2561 y SDict begin H.R end 2235 2561 a 2235 2625 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.5.7) cvn H.B /ANN pdfmark end 2235 2625 a Fw(:)52 b(the)37 b(m)m(utation)f(at)g(ev)m(ery)j(v)m(ertex)g(is)94 2742 y(an)f(in)m(v)m(olution)f(on)g(the)i(set)f(of)g(righ)m(t-equiv)-5 b(alence)37 b(classes)i(of)e(reduced)i(QP-represen)m(tations)94 2858 y(\(Theorem)544 2858 y SDict begin H.S end 544 2858 a Fw(10.13)766 2794 y SDict begin H.R end 766 2794 a 766 2858 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.10.13) cvn H.B /ANN pdfmark end 766 2858 a Fw(\).)194 2974 y(Some)27 b(examples)h(of)f(QP-represen)m(tations)i(and)f(their)f(m)m(utations)g (are)g(giv)m(en)h(in)f(Section)3537 2974 y SDict begin H.S end 3537 2974 a Fw(11)3634 2910 y SDict begin H.R end 3634 2910 a 3634 2974 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (section.11) cvn H.B /ANN pdfmark end 3634 2974 a Fw(.)94 3090 y(All)35 b(these)i(examples)f(treat)f(quiv)m(ers)j(with) d(three)i(v)m(ertices.)54 b(In)36 b(particular,)g(w)m(e)g(describ)s(e)h (the)94 3207 y(e\013ect)24 b(of)f(m)m(utations)f(on)h(a)g(sp)s(ecial)f (family)f(of)i Fv(b)-5 b(and)25 b(r)-5 b(epr)g(esentations)31 b Fw(coming)21 b(from)h(the)i(theory)94 3323 y(of)33 b(string)f(algebras)f([)889 3323 y SDict begin H.S end 889 3323 a Fw(11)987 3259 y SDict begin H.R end 987 3259 a 987 3323 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.BR) cvn H.B /ANN pdfmark end 987 3323 a Fw(,)1047 3323 y SDict begin H.S end 1047 3323 a Fw(21)1144 3259 y SDict begin H.R end 1144 3259 a 1144 3323 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.F) cvn H.B /ANN pdfmark end 1144 3323 a Fw(].)194 3439 y(The)47 b(concluding)e(Section)1263 3439 y SDict begin H.S end 1263 3439 a Fw(12)1361 3375 y SDict begin H.R end 1361 3375 a 1361 3439 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (section.12) cvn H.B /ANN pdfmark end 1361 3439 a 45 w Fw(con)m(tains)h(some)g(op)s(en)g(problems)e (ab)s(out)i(QPs)g(and)g(their)94 3555 y(represen)m(tations)34 b(that)f(w)m(e)g(\014nd)g(essen)m(tial)g(for)f(b)s(etter)h (understanding)g(of)f(the)h(theory)-8 b(.)194 3671 y(In)36 b(the)g(forthcoming)e(con)m(tin)m(uation)h(of)h(this)f(pap)s(er,)i(w)m (e)g(plan)e(to)g(discuss)i(applications)d(of)94 3788 y(QP-represen)m(tations)28 b(and)f(their)f(m)m(utations)g(to)h(the)g (structure)h(of)e(the)h(corresp)s(onding)g(cluster)94 3904 y(algebras.)94 4030 y SDict begin H.S end 94 4030 a 94 4030 a SDict begin 14 H.A end 94 4030 a 94 4030 a SDict begin [ /View [/XYZ H.V] /Dest (section.2) cvn H.B /DEST pdfmark end 94 4030 a 1128 4127 a Fw(2.)49 b Fx(Quivers)37 b(and)h(p)-7 b(a)g(th)38 b(algebras)194 4301 y Fw(A)h Fv(quiver)48 b Fu(Q)38 b Fw(=)f(\()p Fu(Q)955 4316 y Fp(0)995 4301 y Fu(;)17 b(Q)1116 4316 y Fp(1)1156 4301 y Fu(;)g(h;)g(t)p Fw(\))38 b(consists)h(of)e(a)i(pair)e(of)h(\014nite)g (sets)h Fu(Q)2831 4316 y 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5297 y Fq(d)2729 5282 y Fw(\))d(=)h Fu(h)p Fw(\()p Fu(a)3061 5297 y Fq(d)p Fp(+1)3192 5282 y Fw(\),)i(in)e(whic)m(h)94 5398 y(case)42 b(the)f(pro)s(duct)g(is)f (giv)m(en)g(b)m(y)i(concatenation)e(of)g(paths.)68 b(This)41 b(description)f(implies)e(the)94 5514 y(follo)m(wing:)94 5671 y(\(2.1\))528 5671 y SDict begin H.S end 528 5671 a 528 5671 a SDict begin 14 H.A end 528 5671 a 528 5671 a SDict begin [ /View [/XYZ H.V] /Dest (equation.2.1) cvn H.B /DEST pdfmark end 528 5671 a Fw(If)33 b(0)27 b Fr(6)p Fw(=)h Fu(p)f Fr(2)h Fu(A)1049 5630 y Fq(k)1092 5671 y Fu(e)1137 5686 y Fq(i)1198 5671 y Fw(and)33 b(0)27 b Fr(6)p Fw(=)h Fu(q)j Fr(2)d Fu(e)1781 5686 y Fq(i)1810 5671 y Fu(A)1883 5630 y Fq(`)1948 5671 y Fw(for)k(some)h(v)m(ertex)h Fu(i)f Fw(then)g Fu(pq)e Fr(6)p Fw(=)d(0.)94 5737 y SDict begin H.S end 94 5737 a 94 5737 a SDict begin 14 H.A end 94 5737 a 94 5737 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.2.2) cvn H.B /DEST pdfmark end 94 5737 a eop end end %%Page: 7 7 TeXDict begin 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end 94 3303 a 94 3303 a SDict begin 14 H.A end 94 3303 a 94 3303 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.2.3) cvn H.B /DEST pdfmark end 94 3303 a 77 x FA(De\014nition)37 b(2.3.)k Fw(The)34 b Fv(c)-5 b(omplete)34 b(p)-5 b(ath)35 b(algebr)-5 b(a)39 b Fw(of)32 b Fu(Q)h Fw(is)f(de\014ned)i(as)1518 3623 y Fu(R)q Fr(hh)p Fu(A)p Fr(ii)27 b Fw(=)1981 3498 y Fn(1)1953 3528 y Fm(Y)1953 3740 y Fq(d)p Fp(=0)2097 3623 y Fu(A)2170 3582 y Fq(d)2210 3623 y Fu(:)94 3885 y Fw(Th)m(us,)46 b(the)c(elemen)m(ts)g(of)f Fu(R)q Fr(hh)p Fu(A)p Fr(ii)f Fw(are)i(\(p)s(ossibly)f(in\014nite\))f Fu(K)7 b Fw(-linear)40 b(com)m(binations)g(of)h(the)94 4001 y(elemen)m(ts)28 b(of)f(a)g(path)g(basis)h(in)e Fu(R)q Fr(h)p Fu(A)p Fr(i)p Fw(;)j(and)e(the)h(m)m(ultiplication)23 b(in)j Fu(R)q Fr(hh)p Fu(A)p Fr(ii)h Fw(naturally)e(extends)94 4117 y(the)33 b(m)m(ultiplication)28 b(in)k Fu(R)q Fr(h)p Fu(A)p Fr(i)p Fw(.)194 4298 y(Note)25 b(that,)h(if)e(the)h(quiv)m(er)g Fu(Q)g Fw(is)g Fv(acyclic)k 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1889 4810 V 1887 4809 V 1885 4808 V 1883 4807 V 1881 4806 V 1879 4805 V 1877 4804 V 1876 4803 V 1874 4802 V 1872 4802 V 1870 4801 V 1868 4800 V 1867 4800 V 1865 4799 V 1863 4798 V 1862 4798 V 1860 4797 V 1858 4797 V 1857 4797 V 1853 4796 V 1850 4796 V 1847 4795 V 1844 4795 V 1841 4795 V 1838 4795 V 1836 4795 V 1833 4795 V 1830 4796 V 1828 4796 V 1825 4797 V 1823 4798 V 1821 4799 V 1819 4800 V 1817 4801 V 1814 4802 V 1812 4803 V 1811 4805 V 1809 4806 V 1807 4808 V 1805 4810 V 1804 4811 V 1802 4813 V 1801 4815 V 1799 4817 V 1798 4819 V 1797 4821 V 1796 4823 V 1795 4825 V 1794 4827 V 1793 4829 V 1792 4832 V 1791 4834 V 1791 4836 V 1790 4839 V 1790 4841 V 1789 4844 V 1789 4846 V 1789 4848 V 1788 4851 V 1788 4853 V 1788 4856 V 1788 4858 V 1788 4861 V 1789 4863 V 1789 4866 V 1789 4868 V 1790 4870 V 1790 4873 V 1791 4875 V 1791 4878 V 1792 4880 V 1793 4882 V 1794 4884 V 1795 4887 V 1796 4889 V 1797 4891 V 1798 4893 V 1799 4895 V 1801 4897 V 1802 4899 V 1804 4901 V 1805 4902 V 1807 4904 V 1809 4906 V 1811 4907 V 1812 4908 V 1814 4910 V 1817 4911 V 1819 4912 V 1821 4913 V 1823 4914 V 1825 4915 V 1828 4915 V 1830 4916 V 1833 4916 V 1836 4917 V 1838 4917 V 1841 4917 V 1844 4917 V 1847 4917 V 1850 4916 V 1853 4916 V 1857 4915 V 1858 4915 V 1860 4914 V 1862 4914 V 1863 4913 V 1865 4913 V 1867 4912 V 1868 4912 V 1870 4911 V 1872 4910 V 1874 4910 V 1876 4909 V 1877 4908 V 1879 4907 V 1881 4906 V 1883 4905 V 1885 4904 V 1887 4903 V 1889 4902 V 1891 4901 V 1893 4900 V 2034 4879 a Fu(:)94 5057 y Fw(In)44 b(this)e(case)i Fu(R)i Fw(=)f Fu(K)975 5021 y Fq(Q)1031 5030 y Fo(0)1115 5057 y Fw(=)g Fu(K)7 b Fw(,)45 b(and)e Fu(A)j Fw(=)f Fu(K)1928 5021 y Fq(Q)1984 5030 y Fo(1)2067 5057 y Fw(=)g Fu(K)7 b(a)p Fw(.)75 b(W)-8 b(e)43 b(ha)m(v)m(e)h Fu(R)q Fr(h)p Fu(A)p Fr(i)h Fw(=)g Fu(K)7 b Fw([)p Fu(a)p Fw(],)46 b(and)94 5173 y Fu(R)q Fr(hh)p Fu(A)p Fr(ii)27 b Fw(=)h Fu(K)7 b Fw([[)p Fu(a)p Fw(]],)33 b(the)g(algebra)f(of)g(formal)e(p)s(o)m(w)m(er)j(series.)194 5354 y(Let)g Fl(m)28 b Fw(=)f Fl(m)p 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Fw(ha)m(ving)h(zero)h(in)m(tersec-)94 2281 y(tion)g(with)g Fu(R)f Fw(=)f Fu(A)817 2245 y Fp(0)856 2281 y Fw(.)55 b(Indeed,)39 b(it)c(is)h(enough)g(to)g(sho)m(w)i(that)e(an)m(y)h (elemen)m(t)f Fu(x)f Fr(2)f Fu(R)q Fr(hh)p Fu(A)p Fr(ii)24 b(\000)h Fl(m)94 2397 y Fw(generates)47 b(an)f(ideal)e(ha)m(ving)i (nonzero)g(in)m(tersection)g(with)f Fu(R)q Fw(.)84 b(Let)46 b Fu(x)k Fw(=)g Fu(c)32 b Fw(+)f Fu(y)48 b Fw(with)e Fu(c)g Fw(a)94 2514 y(nonzero)34 b(elemen)m(t)e(of)h Fu(R)q Fw(,)g(and)g Fu(y)e Fr(2)d Fl(m)p Fw(.)45 b(Multiplying)30 b Fu(x)j Fw(on)g(b)s(oth)f(sides)i(b)m(y)f(suitable)f(elemen)m(ts)94 2630 y(of)41 b Fu(R)q Fw(,)i(w)m(e)f(can)f(assume)g(that)g Fu(c)g Fw(=)h Fu(e)1509 2645 y Fq(i)1578 2630 y Fw(for)e(some)h Fu(i)h Fr(2)g Fu(Q)2248 2645 y Fp(0)2288 2630 y Fw(,)h(and)e Fu(e)2601 2645 y Fq(i)2629 2630 y Fu(y)k Fw(=)d Fu(y)t(e)2937 2645 y Fq(i)3006 2630 y Fw(=)f Fu(y)t Fw(.)68 b(But)41 b(then)94 2746 y Fu(z)33 b Fw(=)27 b Fu(e)320 2761 y Fq(i)370 2746 y Fr(\000)22 b Fu(y)i Fw(+)d Fu(y)690 2710 y Fp(2)750 2746 y Fr(\000)h Fu(y)901 2710 y Fp(3)961 2746 y Fw(+)f Fr(\001)c(\001)g(\001)47 b Fw(is)31 b(a)h(w)m (ell-de\014ned)h(elemen)m(t)f(of)f Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fw(,)h(and)g(w)m(e)h(ha)m(v)m(e)h Fu(xz)e Fw(=)c Fu(e)3606 2761 y Fq(i)3634 2746 y Fw(,)94 2862 y(pro)m(ving)33 b(our)f(claim.)194 2979 y(This)k(c)m(haracterization)e(of)h Fl(m)g Fw(implies)e(that)i(it)g(is)f(in)m(v)-5 b(arian)m(t)34 b(under)i(an)m(y)g Fu(R)q Fw(-algebra)e(auto-)94 3095 y(morphism)i Fu(')g Fw(of)h Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fw(.)56 b(Th)m(us,)38 b Fu(')f Fw(is)g(con)m(tin)m(uous,)i(i.e.,)e(is)g (an)g(automorphism)e(of)h Fu(R)q Fr(hh)p Fu(A)p Fr(ii)94 3211 y Fw(as)d(a)g(top)s(ological)28 b(algebra.)194 3327 y(The)i(same)f(argumen)m(t)g(sho)m(ws)i(that,)e(more)g(generally)-8 b(,)29 b(if)f Fu(A)h Fw(and)g Fu(A)2696 3291 y Fn(0)2748 3327 y Fw(are)h(\014nite-dimensional)94 3444 y Fu(R)q Fw(-bimo)s(dules)39 b(then)i(an)m(y)g(homomorphism)d(of)i Fu(R)q Fw(-algebras)f Fu(')i Fw(:)h Fu(R)q Fr(hh)p Fu(A)p Fr(ii)e(!)h Fu(R)q Fr(hh)p Fu(A)3292 3407 y Fn(0)3315 3444 y Fr(ii)f Fw(sends)94 3560 y Fl(m)p Fw(\()p Fu(A)p Fw(\))30 b(in)m(to)f Fl(m)p Fw(\()p Fu(A)731 3524 y Fn(0)755 3560 y Fw(\),)h(hence)h(is)e(con)m(tin)m(uous.)43 b(Th)m(us,)32 b Fu(')d Fw(is)g(uniquely)h(determined)f(b)m(y)i(its)e(restric-)94 3676 y(tion)34 b(to)h Fu(A)492 3640 y Fp(1)564 3676 y Fw(=)c Fu(A)p Fw(,)36 b(whic)m(h)g(is)e(a)h Fu(R)q Fw(-bimo)s(dule)e (homomorphism)f Fu(A)g Fr(!)f Fl(m)p Fw(\()p Fu(A)2923 3640 y Fn(0)2947 3676 y Fw(\))h(=)g Fu(A)3198 3640 y Fn(0)3245 3676 y Fr(\010)24 b Fl(m)p Fw(\()p Fu(A)3533 3640 y Fn(0)3557 3676 y Fw(\))3595 3640 y Fp(2)3634 3676 y Fw(.)94 3795 y(W)-8 b(e)43 b(write)f Fu(')p Fr(j)623 3810 y Fq(A)724 3795 y Fw(=)i(\()p Fu(')946 3759 y Fp(\(1\))1040 3795 y Fu(;)17 b(')1148 3759 y Fp(\(2\))1242 3795 y Fw(\),)45 b(where)e Fu(')1707 3759 y Fp(\(1\))1846 3795 y Fw(:)h Fu(A)g Fr(!)g Fu(A)2251 3759 y Fn(0)2317 3795 y Fw(and)e Fu(')2580 3759 y Fp(\(2\))2719 3795 y Fw(:)i Fu(A)g Fr(!)g Fl(m)p Fw(\()p Fu(A)3238 3759 y Fn(0)3262 3795 y Fw(\))3300 3759 y Fp(2)3382 3795 y Fw(are)e Fu(R)q Fw(-)94 3911 y(bimo)s(dule)31 b(homomorphisms.)94 3931 y SDict begin H.S end 94 3931 a 94 3931 a SDict begin 14 H.A end 94 3931 a 94 3931 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.2.5) cvn H.B /DEST pdfmark end 94 3931 a 167 x FA(Prop)s(osition)43 b(2.5.)i Fv(A)n(ny)40 b(p)-5 b(air)40 b Fw(\()p Fu(')1446 4061 y Fp(\(1\))1540 4098 y Fu(;)17 b(')1648 4061 y Fp(\(2\))1742 4098 y Fw(\))40 b Fv(of)g Fu(R)q Fv(-bimo)-5 b(dule)39 b(homomorphisms)f Fu(')3254 4061 y Fp(\(1\))3386 4098 y Fw(:)g Fu(A)g Fr(!)94 4217 y Fu(A)167 4181 y Fn(0)233 4217 y Fv(and)j Fu(')493 4181 y Fp(\(2\))628 4217 y Fw(:)g Fu(A)g Fr(!)f Fl(m)p Fw(\()p Fu(A)1137 4181 y Fn(0)1161 4217 y Fw(\))1199 4181 y Fp(2)1281 4217 y Fv(gives)h(rise)g(to)i(a)e (unique)h(homomorphism)e(of)i(top)-5 b(olo)g(gic)g(al)40 b Fu(R)q Fv(-)94 4336 y(algebr)-5 b(as)40 b Fu(')e Fw(:)g Fu(R)q Fr(hh)p Fu(A)p Fr(ii)f(!)h Fu(R)q Fr(hh)p Fu(A)1342 4300 y Fn(0)1365 4336 y Fr(ii)i Fv(such)g(that)h Fu(')p Fr(j)2007 4351 y Fq(A)2101 4336 y Fw(=)d(\()p Fu(')2317 4300 y Fp(\(1\))2411 4336 y Fu(;)17 b(')2519 4300 y Fp(\(2\))2613 4336 y Fw(\))p Fv(.)61 b(F)-7 b(urthermor)i(e,)41 b Fu(')g Fv(is)f(an)94 4455 y(isomorphism)34 b(if)g(and)h(only)f(if)h Fu(')1323 4419 y Fp(\(1\))1452 4455 y Fv(is)g(a)f Fu(R)q Fv(-bimo)-5 b(dule)34 b(isomorphism)f Fu(A)28 b Fr(!)f Fu(A)3032 4419 y Fn(0)3056 4455 y Fv(.)94 4641 y(Pr)-5 b(o)g(of.)42 b Fw(The)32 b(uniqueness)g(of)f Fu(')g Fw(is)f(clear.)43 b(T)-8 b(o)31 b(sho)m(w)h(the)f(existence,)i(let)d Fu(Q)2858 4656 y Fp(1)2925 4641 y Fw(=)e Fr(f)p Fu(a)3130 4656 y Fp(1)3169 4641 y Fu(;)17 b(:)g(:)g(:)f(;)h(a)3439 4656 y Fq(N)3506 4641 y Fr(g)28 b(\032)94 4757 y Fu(A)41 b Fw(=)e Fu(A)396 4721 y Fp(1)476 4757 y Fw(b)s(e)h(the)g(set)g(of)g (arro)m(ws)g(in)f Fu(A)p Fw(.)65 b(W)-8 b(riting)38 b(an)i(elemen)m(t)f Fu(x)h Fr(2)h Fu(R)q Fr(hh)p Fu(A)p Fr(ii)e Fw(as)h(an)f(in\014nite)94 4874 y Fu(K)7 b Fw(-linear)36 b(com)m(bination)f(of)i(the)g(elemen)m (ts)h(of)f(the)g(corresp)s(onding)g(path)h(basis)f(in)f Fu(R)q Fr(h)p Fu(A)p Fr(i)p Fw(,)i(w)m(e)94 4990 y(express)j Fu(x)e Fw(as)f(a)g(\(non-comm)m(utativ)m(e\))f(formal)f(p)s(o)m(w)m(er) j(series)g Fu(F)14 b Fw(\()p Fu(a)2635 5005 y Fp(1)2674 4990 y Fu(;)j(:)g(:)g(:)f(;)h(a)2944 5005 y Fq(N)3011 4990 y Fw(\).)61 b(Therefore,)41 b(a)94 5109 y(desired)e Fu(R)q Fw(-algebra)e(homomorphism)e(can)j(b)s(e)g(obtained)g(b)m(y)h (setting)e Fu(')p Fw(\()p Fu(x)p Fw(\))h(=)e Fu(F)14 b Fw(\()p Fu(')3299 5073 y Fp(\(1\))3393 5109 y Fw(\()p Fu(a)3482 5124 y Fp(1)3522 5109 y Fw(\))26 b(+)94 5228 y Fu(')158 5192 y Fp(\(2\))253 5228 y Fw(\()p Fu(a)342 5243 y Fp(1)381 5228 y Fw(\))p Fu(;)17 b(:)g(:)g(:)f(;)h(')702 5192 y Fp(\(1\))796 5228 y Fw(\()p Fu(a)885 5243 y Fq(N)952 5228 y Fw(\))22 b(+)g Fu(')1174 5192 y Fp(\(2\))1269 5228 y Fw(\()p Fu(a)1358 5243 y Fq(N)1425 5228 y Fw(\)\).)194 5347 y(If)49 b Fu(')h Fw(is)f(an)g(isomorphism)e(then)j Fu(')1579 5311 y Fp(\(1\))1729 5347 y Fw(:)56 b Fu(A)h Fr(!)f Fu(A)2171 5311 y Fn(0)2243 5347 y Fw(is)49 b(clearly)g(an)g (isomorphism)e(of)i Fu(R)q Fw(-)94 5464 y(bimo)s(dules.)67 b(T)-8 b(o)41 b(sho)m(w)i(the)e(con)m(v)m(erse)i(implication,)c(w)m(e)j (can)g(iden)m(tify)e Fu(A)h Fw(and)g Fu(A)3231 5428 y Fn(0)3295 5464 y Fw(with)g(the)94 5580 y(help)36 b(of)g Fu(')488 5544 y Fp(\(1\))582 5580 y Fw(,)h(and)g(so)f(assume)g(that)g Fu(')1582 5544 y Fp(\(1\))1713 5580 y Fw(is)f(the)i(iden)m(tit)m(y)f (automorphism)e(of)h Fu(A)p Fw(.)55 b(Then)37 b(the)94 5696 y(\(in\014nite\))42 b(matrix)f(that)h(expresses)k Fu(')c Fw(as)h(a)f Fu(K)7 b Fw(-linear)41 b(map)g(in)h(the)h(path)g (basis)f(of)g Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p eop end end %%Page: 9 9 TeXDict begin HPSdict begin 9 8 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.9) cvn H.B /DEST pdfmark end 94 218 a 1179 w Fs(QUIVERS)31 b(WITH)i(POTENTIALS)g(I)1140 b(9)94 417 y Fw(is)38 b(lo)m(w)m(er-triangular)e(with)i(all)f(the)i(diagonal)d(en) m(tries)j(equal)f(to)g(1)h(\(w)m(e)g(order)g(the)g(basis)f(ele-)94 533 y(men)m(ts)j(so)g(that)f(their)f(degrees)j(w)m(eakly)f(increase\).) 67 b(Clearly)-8 b(,)41 b(suc)m(h)h(a)e(matrix)e(is)i(in)m(v)m(ertible,) 94 649 y(completing)31 b(the)i(pro)s(of)f(of)g(Prop)s(osition)1652 649 y SDict begin H.S end 1652 649 a Fw(2.5)1776 585 y SDict begin H.R end 1776 585 a 1776 649 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.2.5) cvn H.B /ANN pdfmark end 1776 649 a Fw(.)1781 b Fh(\003)94 725 y SDict begin H.S end 94 725 a 94 725 a SDict begin 14 H.A end 94 725 a 94 725 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.2.6) cvn H.B /DEST pdfmark end 94 725 a 102 x FA(De\014nition)53 b(2.6.)c Fw(Let)f Fu(')f Fw(b)s(e)g(the)g(automorphism)e(of)i Fu(R)q Fr(hh)p Fu(A)p Fr(ii)f Fw(corresp)s(onding)h(to)g(a)f(pair)94 946 y(\()p Fu(')196 910 y Fp(\(1\))291 946 y Fu(;)17 b(')399 910 y Fp(\(2\))493 946 y Fw(\))24 b(as)g(in)g(Prop)s(osition) 1288 946 y SDict begin H.S end 1288 946 a Fw(2.5)1413 882 y SDict begin H.R end 1413 882 a 1413 946 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.2.5) cvn H.B /ANN pdfmark end 1413 946 a Fw(.)41 b(If)24 b Fu(')1634 910 y Fp(\(2\))1756 946 y Fw(=)j(0,)f(then)f(w)m(e)g(call)e Fu(')h Fw(a)g Fv(change)i(of)h(arr)-5 b(ows)p Fw(.)40 b(If)24 b Fu(')3567 910 y Fp(\(1\))94 1062 y Fw(is)33 b(the)h(iden)m(tit)m(y)f (automorphism)f(of)h Fu(A)p Fw(,)g(w)m(e)i(sa)m(y)f(that)f Fu(')g Fw(is)g(a)g Fv(unitriangular)44 b Fw(automorphism;)94 1178 y(furthermore,)33 b(w)m(e)g(sa)m(y)h(that)e Fu(')h Fw(is)f(of)g Fv(depth)40 b Fu(d)27 b Fr(\025)h Fw(1,)k(if)g Fu(')2202 1142 y Fp(\(2\))2296 1178 y Fw(\()p Fu(A)p Fw(\))c Fr(\032)g Fl(m)2654 1142 y Fq(d)p Fp(+1)2785 1178 y Fw(.)194 1355 y(The)48 b(follo)m(wing)c(prop)s(ert)m(y)k(of)f (unitriangular)d(automorphisms)h(is)i(immediate)d(from)i(the)94 1472 y(de\014nitions:)590 1634 y(If)33 b Fu(')f Fw(is)g(an)h (unitriangular)c(automorphism)i(of)h Fu(R)q Fr(hh)p Fu(A)p Fr(ii)g Fw(of)g(depth)h Fu(d)p Fw(,)94 1634 y SDict begin H.S end 94 1634 a 94 1634 a SDict begin 17.5 H.A end 94 1634 a 94 1634 a SDict begin [ /View [/XYZ H.V] /Dest (equation.2.4) cvn H.B /DEST pdfmark end 94 1634 a Fw(\(2.4\))590 1797 y(then)g Fu(')p Fw(\()p Fu(u)p Fw(\))22 b Fr(\000)g Fu(u)28 b Fr(2)g Fl(m)1383 1756 y Fq(n)p Fp(+)p Fq(d)1554 1797 y Fw(for)k Fu(u)27 b Fr(2)h Fl(m)1956 1756 y Fq(n)2003 1797 y Fw(.)94 1932 y SDict begin H.S end 94 1932 a 94 1932 a SDict begin 14 H.A end 94 1932 a 94 1932 a SDict begin [ /View [/XYZ H.V] /Dest (section.3) cvn H.B /DEST pdfmark end 94 1932 a 859 2009 a Fw(3.)48 b Fx(Potentials)38 b(and)g(their)f(Ja)n(cobian)h(ideals)194 2183 y Fw(In)k(this)e(section) i(w)m(e)g(in)m(tro)s(duce)f(some)g(of)g(our)g(main)e(ob)5 b(jects)43 b(of)d(study:)62 b(p)s(oten)m(tials)40 b(and)94 2300 y(their)e(Jacobian)e(ideals)h(in)g(the)h(complete)f(path)g (algebra)g Fu(R)q Fr(hh)p Fu(A)p Fr(ii)g Fw(giv)m(en)g(b)m(y)i (De\014nition)3510 2300 y SDict begin H.S end 3510 2300 a Fw(2.3)3634 2235 y SDict begin H.R end 3634 2235 a 3634 2300 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.2.3) cvn H.B /ANN pdfmark end 3634 2300 a Fw(.)94 2416 y(W)-8 b(e)43 b(\014x)f(a)f(path)h(basis)f(in)g Fu(R)q Fr(h)p Fu(A)p Fr(i)p Fw(;)46 b(recall)40 b(that)i(it)e(consists)j(of)e(the)h (elemen)m(ts)g Fu(e)3131 2431 y Fq(i)3202 2416 y Fr(2)i Fu(R)g Fw(=)f Fu(A)3622 2380 y Fp(0)94 2532 y Fw(together)25 b(with)e(the)i(pro)s(ducts)g Fu(a)1291 2547 y Fp(1)1347 2532 y Fr(\001)17 b(\001)g(\001)e Fu(a)1531 2547 y Fq(d)1596 2532 y Fw(\(paths\))24 b(suc)m(h)i(that)d(all)f Fu(a)2521 2547 y Fq(k)2588 2532 y Fw(b)s(elong)h(to)h Fu(Q)3079 2547 y Fp(1)3119 2532 y Fw(,)i(and)e Fu(t)p Fw(\()p Fu(a)3477 2547 y Fq(k)3520 2532 y Fw(\))k(=)94 2648 y Fu(h)p Fw(\()p Fu(a)239 2663 y Fq(k)r Fp(+1)373 2648 y Fw(\))33 b(for)h(1)29 b Fr(\024)i Fu(k)i(<)c(d)p Fw(.)47 b(Then)35 b(eac)m(h)g(space)g Fu(A)1907 2612 y Fq(d)1981 2648 y Fw(has)g(a)e(direct)h Fu(R)q Fw(-bimo)s(dule)d(decomp)s(osition)94 2764 y Fu(A)167 2728 y Fq(d)239 2764 y Fw(=)346 2690 y Fm(L)457 2794 y Fq(i;j)t Fn(2)p Fq(Q)637 2803 y Fo(0)691 2764 y Fu(A)764 2728 y Fq(d)764 2789 y(i;j)844 2764 y Fw(,)k(where)h(the)f(comp)s(onen) m(t)f Fu(A)1933 2728 y Fq(d)1933 2789 y(i;j)2048 2764 y Fw(is)g(spanned)i(b)m(y)f(the)g(paths)g Fu(a)3156 2779 y Fp(1)3213 2764 y Fr(\001)17 b(\001)g(\001)e Fu(a)3397 2779 y Fq(d)3472 2764 y Fw(with)94 2889 y Fu(h)p Fw(\()p Fu(a)239 2904 y Fp(1)279 2889 y Fw(\))28 b(=)f Fu(i)33 b Fw(and)g Fu(t)p Fw(\()p Fu(a)828 2904 y Fq(d)869 2889 y Fw(\))27 b(=)h Fu(j)6 b Fw(.)94 2914 y SDict begin H.S end 94 2914 a 94 2914 a SDict begin 14 H.A end 94 2914 a 94 2914 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.3.1) cvn H.B /DEST pdfmark end 94 2914 a 153 x FA(De\014nition)37 b(3.1.)318 3206 y Fr(\017)k Fw(F)-8 b(or)28 b(eac)m(h)i Fu(d)d Fr(\025)h Fw(1,)i(w)m(e)g(de\014ne)g(the)f Fv(cyclic)i(p)-5 b(art)39 b Fw(of)28 b Fu(A)2322 3170 y Fq(d)2392 3206 y Fw(as)h(the)g(sub-)p Fu(R)q Fw(-bimo)s(dule)e Fu(A)3454 3170 y Fq(d)3454 3230 y Fp(cyc)3586 3206 y Fw(=)409 3259 y Fm(L)520 3363 y Fq(i)p Fn(2)p Fq(Q)647 3372 y Fo(0)702 3334 y Fu(A)775 3298 y Fq(d)775 3358 y(i;i)847 3334 y Fw(.)58 b(Th)m(us,)41 b Fu(A)1286 3298 y Fq(d)1286 3358 y Fp(cyc)1427 3334 y Fw(is)c(the)h(span)h(of)e(all)e(paths)j Fu(a)2515 3349 y Fp(1)2571 3334 y Fr(\001)17 b(\001)g(\001)e Fu(a)2755 3349 y Fq(d)2833 3334 y Fw(with)38 b Fu(h)p Fw(\()p Fu(a)3206 3349 y Fp(1)3245 3334 y Fw(\))f(=)f Fu(t)p Fw(\()p Fu(a)3556 3349 y Fq(d)3596 3334 y Fw(\);)409 3453 y(w)m(e)e(call)d(suc)m(h)j(paths)f Fv(cyclic)p Fw(.)318 3570 y Fr(\017)41 b Fw(W)-8 b(e)33 b(de\014ne)h(a)e(closed)h(v)m(ector) h(subspace)g Fu(R)q Fr(hh)p Fu(A)p Fr(ii)2232 3585 y Fp(cyc)2363 3570 y Fr(\022)28 b Fu(R)q Fr(hh)p Fu(A)p Fr(ii)k Fw(b)m(y)h(setting)1435 3814 y Fu(R)q Fr(hh)p Fu(A)p Fr(ii)1739 3829 y Fp(cyc)1870 3814 y Fw(=)2001 3690 y Fn(1)1973 3720 y Fm(Y)1973 3932 y Fq(d)p Fp(=1)2117 3814 y Fu(A)2190 3773 y Fq(d)2190 3839 y Fp(cyc)2294 3814 y Fu(;)409 4071 y Fw(and)g(call)e(the)i(elemen)m(ts)f(of)h Fu(R)q Fr(hh)p Fu(A)p Fr(ii)1760 4086 y Fp(cyc)1895 4071 y Fv(p)-5 b(otentials)p Fw(.)318 4187 y Fr(\017)41 b Fw(F)-8 b(or)24 b(ev)m(ery)i Fu(\030)32 b Fr(2)c Fu(A)1067 4151 y Fq(?)1106 4187 y Fw(,)e(w)m(e)g(de\014ne)f(the)g Fv(cyclic)i(derivative)k Fu(@)2487 4202 y Fq(\030)2549 4187 y Fw(as)25 b(the)f(con)m(tin)m(uous)h Fu(K)7 b Fw(-linear)409 4303 y(map)32 b Fu(R)q Fr(hh)p Fu(A)p Fr(ii)930 4318 y Fp(cyc)1061 4303 y Fr(!)27 b Fu(R)q Fr(hh)p Fu(A)p Fr(ii)32 b Fw(acting)f(on)i(paths)g(b)m(y)94 4565 y(\(3.1\))937 4565 y SDict begin H.S end 937 4565 a 937 4565 a SDict begin 14 H.A end 937 4565 a 937 4565 a SDict begin [ /View [/XYZ H.V] /Dest (equation.3.1) cvn H.B /DEST pdfmark end 937 4565 a Fu(@)988 4580 y Fq(\030)1026 4565 y Fw(\()p Fu(a)1115 4580 y Fp(1)1171 4565 y Fr(\001)17 b(\001)g(\001)e Fu(a)1355 4580 y Fq(d)1396 4565 y Fw(\))27 b(=)1619 4441 y Fq(d)1565 4471 y Fm(X)1573 4683 y Fq(k)r Fp(=1)1725 4565 y Fu(\030)5 b Fw(\()p Fu(a)1862 4580 y Fq(k)1905 4565 y Fw(\))p Fu(a)1994 4580 y Fq(k)r Fp(+1)2143 4565 y Fr(\001)17 b(\001)g(\001)e Fu(a)2327 4580 y Fq(d)2368 4565 y Fu(a)2419 4580 y Fp(1)2475 4565 y Fr(\001)i(\001)g(\001)e Fu(a)2659 4580 y Fq(k)r Fn(\000)p Fp(1)2792 4565 y Fu(:)318 4822 y Fr(\017)41 b Fw(F)-8 b(or)31 b(ev)m(ery)i(p)s(oten)m(tial)c Fu(S)6 b Fw(,)32 b(w)m(e)g(de\014ne)g(its)f Fv(Jac)-5 b(obian)33 b(ide)-5 b(al)41 b Fu(J)9 b Fw(\()p Fu(S)d Fw(\))31 b(as)g(the)h (closure)f(of)g(the)409 4938 y(\(t)m(w)m(o-sided\))39 b(ideal)e(in)g Fu(R)q Fr(hh)p Fu(A)p Fr(ii)h Fw(generated)h(b)m(y)g (the)g(elemen)m(ts)g Fu(@)2846 4953 y Fq(\030)2884 4938 y Fw(\()p Fu(S)6 b Fw(\))38 b(for)g(all)e Fu(\030)42 b Fr(2)c Fu(A)3622 4902 y Fq(?)409 5054 y Fw(\(see)c(\()643 5054 y SDict begin H.S end 643 5054 a Fw(2.3)767 4990 y SDict begin H.R end 767 4990 a 767 5054 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.2.3) cvn H.B /ANN pdfmark end 767 5054 a Fw(\)\);)f(clearly)-8 b(,)31 b Fu(J)9 b Fw(\()p Fu(S)d Fw(\))33 b(is)f(a)g(t)m(w)m(o-sided)h(ideal)e(in)h Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fw(.)318 5170 y Fr(\017)41 b Fw(W)-8 b(e)37 b(call)e(the)i(quotien)m(t)g Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fu(=J)9 b Fw(\()p Fu(S)d Fw(\))35 b(the)j Fv(Jac)-5 b(obian)37 b(algebr)-5 b(a)43 b Fw(of)37 b Fu(S)6 b Fw(,)37 b(and)g(denote)g(it)409 5287 y(b)m(y)d Fr(P)8 b Fw(\()p Fu(Q;)17 b(S)6 b Fw(\))32 b(or)g Fr(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\).)194 5464 y(An)28 b(easy)h(c)m(hec)m (k)h(sho)m(ws)g(that)e(a)f(cyclic)h(deriv)-5 b(ativ)m(e)27 b Fu(@)2133 5479 y Fq(\030)2199 5464 y Fw(:)h Fu(R)q Fr(hh)p Fu(A)p Fr(ii)2558 5479 y Fp(cyc)2689 5464 y Fr(!)f Fu(R)q Fr(hh)p Fu(A)p Fr(ii)g Fw(do)s(es)i(not)e(de-)94 5580 y(p)s(end)g(on)f(the)g(c)m(hoice)g(of)g(a)f(path)h(basis.)41 b(F)-8 b(urthermore,)27 b(cyclic)e(deriv)-5 b(ativ)m(es)26 b(do)g(not)g(distinguish)94 5696 y(b)s(et)m(w)m(een)35 b(the)e(p)s(oten)m(tials)e(that)i(are)f(equiv)-5 b(alen)m(t)33 b(in)f(the)h(follo)m(wing)c(sense.)94 5716 y SDict begin H.S end 94 5716 a 94 5716 a SDict begin 14 H.A end 94 5716 a 94 5716 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.3.2) cvn H.B /DEST pdfmark end 94 5716 a eop end end %%Page: 10 10 TeXDict begin HPSdict begin 10 9 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.10) cvn H.B /DEST pdfmark end 94 218 a Fs(10)492 b(HARM)25 b(DERKSEN,)f(JERZY)i(WEYMAN,)f(AND)f(ANDREI)g(ZELEVINSKY)94 417 y FA(De\014nition)30 b(3.2.)37 b Fw(Tw)m(o)27 b(p)s(oten)m(tials)e Fu(S)33 b Fw(and)26 b Fu(S)1821 381 y Fn(0)1871 417 y Fw(are)g Fv(cyclic)-5 b(al)5 b(ly)29 b(e)-5 b(quivalent)35 b Fw(if)26 b Fu(S)15 b Fr(\000)10 b Fu(S)3206 381 y Fn(0)3256 417 y Fw(lies)25 b(in)h(the)94 533 y(closure)35 b(of)g(the)g(span)g(of) g(all)d(elemen)m(ts)j(of)g(the)g(form)e Fu(a)2155 548 y Fp(1)2212 533 y Fr(\001)17 b(\001)g(\001)d Fu(a)2395 548 y Fq(d)2460 533 y Fr(\000)24 b Fu(a)2612 548 y Fp(2)2668 533 y Fr(\001)17 b(\001)g(\001)e Fu(a)2852 548 y Fq(d)2893 533 y Fu(a)2944 548 y Fp(1)2983 533 y Fw(,)36 b(where)g Fu(a)3381 548 y Fp(1)3437 533 y Fr(\001)17 b(\001)g(\001)e Fu(a)3621 548 y Fq(d)94 649 y Fw(is)33 b(a)f(cyclic)g(path.)194 824 y(The)i(follo)m(wing)c(prop)s(osition)g(is)i(immediate)e(from)i(\() 2163 824 y SDict begin H.S end 2163 824 a Fw(3.1)2287 760 y SDict begin H.R end 2287 760 a 2287 824 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.3.1) cvn H.B /ANN pdfmark end 2287 824 a Fw(\).)94 849 y SDict begin H.S end 94 849 a 94 849 a SDict begin 14 H.A end 94 849 a 94 849 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.3.3) cvn H.B /DEST pdfmark end 94 849 a 149 x FA(Prop)s(osition)j(3.3.)41 b Fv(If)34 b(two)g(p)-5 b(otentials)33 b Fu(S)40 b Fv(and)34 b Fu(S)1989 962 y Fn(0)2046 998 y Fv(ar)-5 b(e)34 b(cyclic)-5 b(al)5 b(ly)33 b(e)-5 b(quivalent,)34 b(then)g Fu(@)3378 1013 y Fq(\030)3416 998 y Fw(\()p Fu(S)6 b Fw(\))28 b(=)94 1114 y Fu(@)145 1129 y Fq(\030)184 1114 y Fw(\()p Fu(S)288 1078 y Fn(0)311 1114 y Fw(\))35 b Fv(for)f(al)5 b(l)35 b Fu(\030)d Fr(2)c Fu(A)921 1078 y Fq(?)960 1114 y Fv(,)35 b(henc)-5 b(e)34 b Fu(J)9 b Fw(\()p Fu(S)d Fw(\))28 b(=)f Fu(J)9 b Fw(\()p Fu(S)1797 1078 y Fn(0)1820 1114 y Fw(\))35 b Fv(and)g Fr(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\))27 b(=)h Fr(P)8 b Fw(\()p Fu(A;)17 b(S)2848 1078 y Fn(0)2871 1114 y Fw(\))p Fv(.)194 1289 y Fw(It)32 b(is)f(easy)i(to)e(see)i(that)e (the)h(de\014nition)f(of)g(cyclical)f(equiv)-5 b(alence)32 b(do)s(es)g(not)g(dep)s(end)g(on)g(the)94 1405 y(c)m(hoice)h(of)g(a)f (path)g(basis.)44 b(In)33 b(fact,)f(it)g(can)h(b)s(e)f(giv)m(en)h(in)f (more)g(in)m(v)-5 b(arian)m(t)31 b(terms)i(as)f(follo)m(ws.)94 1424 y SDict begin H.S end 94 1424 a 94 1424 a SDict begin 14 H.A end 94 1424 a 94 1424 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.3.4) cvn H.B /DEST pdfmark end 94 1424 a 155 x FA(De\014nition)e(3.4.)38 b Fw(F)-8 b(or)27 b(an)m(y)h(top)s (ological)23 b Fu(K)7 b Fw(-algebra)26 b Fu(U)10 b Fw(,)29 b(its)e Fv(tr)-5 b(ac)g(e)30 b(sp)-5 b(ac)g(e)34 b Fw(T)-8 b(r\()p Fu(U)10 b Fw(\))28 b(is)e(de\014ned)j(as)94 1695 y(T)-8 b(r)q(\()p Fu(U)10 b Fw(\))28 b(=)f Fu(U)5 b(=)p Fr(f)p Fu(U;)17 b(U)10 b Fr(g)p Fw(,)31 b(where)h Fr(f)p Fu(U;)17 b(U)10 b Fr(g)30 b Fw(is)g(the)h(closure)g(of)f(the)h(v)m (ector)h(subspace)g(in)e Fu(U)41 b Fw(spanned)94 1812 y(b)m(y)34 b(all)c(comm)m(utators.)43 b(W)-8 b(e)33 b(denote)g(b)m(y)h Fu(\031)d Fw(=)d Fu(\031)1855 1827 y Fq(U)1942 1812 y Fw(:)g Fu(U)38 b Fr(!)27 b Fw(T)-8 b(r\()p Fu(U)10 b Fw(\))34 b(the)f(canonical)e(pro)5 b(jection.)194 1986 y(The)34 b(follo)m(wing)c(prop)s(osition)g(is)i(a)h(direct)f (consequence)k(of)c(the)h(de\014nitions.)94 2005 y SDict begin H.S end 94 2005 a 94 2005 a SDict begin 14 H.A end 94 2005 a 94 2005 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.3.5) cvn H.B /DEST pdfmark end 94 2005 a 155 x FA(Prop)s(osition)44 b(3.5.)h Fv(Two)c(p)-5 b(otentials)41 b Fu(S)47 b Fv(and)40 b Fu(S)1966 2124 y Fn(0)2031 2160 y Fv(ar)-5 b(e)41 b(cyclic)-5 b(al)5 b(ly)40 b(e)-5 b(quivalent)41 b(if)g(and)g(only)g(if)94 2276 y Fu(\031)149 2292 y Fq(R)p Fn(hh)p Fq(A)p Fn(ii)370 2276 y Fw(\()p Fu(S)6 b Fw(\))48 b(=)h Fu(\031)740 2292 y Fq(R)p Fn(hh)p Fq(A)p Fn(ii)961 2276 y Fw(\()p Fu(S)1065 2240 y Fn(0)1088 2276 y Fw(\))p Fv(.)78 b(Thus,)49 b(the)d(Jac)-5 b(obian)45 b(ide)-5 b(al)46 b(and)f(the)h(Jac)-5 b(obian)45 b(algebr)-5 b(a)46 b(of)g(a)94 2393 y(p)-5 b(otential)35 b Fu(S)41 b Fv(dep)-5 b(end)33 b(only)i(on)g(the)g(image)e(of)i Fu(S)41 b Fv(in)34 b Fw(T)-8 b(r)q(\()p Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fw(\))p Fv(.)194 2567 y Fw(Recall)25 b(that)h(w)m(e)i(iden)m(tify)e(the)g(set)i (of)e(arro)m(ws)h Fu(Q)1970 2582 y Fp(1)2036 2567 y Fw(with)f(a)g Fu(K)7 b Fw(-basis)26 b(in)g Fu(A)h Fw(=)h Fu(A)3068 2531 y Fp(1)3108 2567 y Fw(.)41 b(F)-8 b(or)26 b Fu(a)i Fr(2)g Fu(Q)3595 2582 y Fp(1)3634 2567 y Fw(,)94 2683 y(w)m(e)k(will)c(use)k(the)f(notation)e Fu(@)1190 2698 y Fq(a)1263 2683 y Fw(for)h(the)h(cyclic)f(deriv)-5 b(ativ)m(e)30 b Fu(@)2338 2698 y Fq(a)2375 2679 y Fg(?)2417 2683 y Fw(,)h(where)g Fu(Q)2831 2647 y Fq(?)2831 2708 y Fp(1)2899 2683 y Fw(=)c Fr(f)p Fu(a)3103 2647 y Fq(?)3170 2683 y Fr(j)h Fu(a)g Fr(2)g Fu(Q)3476 2698 y Fp(1)3515 2683 y Fr(g)j Fw(is)94 2799 y(the)i(dual)f(basis)h(of)f Fu(Q)907 2814 y Fp(1)979 2799 y Fw(in)g Fu(A)1166 2763 y Fq(?)1205 2799 y Fw(.)94 2819 y SDict begin H.S end 94 2819 a 94 2819 a SDict begin 14 H.A end 94 2819 a 94 2819 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.3.6) cvn H.B /DEST pdfmark end 94 2819 a 155 x FA(Example)i(3.6.)41 b Fw(Consider)30 b(the)h(quiv)m(er)g Fu(Q)d Fw(=)g(\()p Fu(Q)1956 2989 y Fp(0)1995 2974 y Fu(;)17 b(Q)2116 2989 y Fp(1)2156 2974 y Fw(\))30 b(with)g Fu(Q)2521 2989 y Fp(0)2588 2974 y Fw(=)e Fr(f)p Fw(1)p Fu(;)17 b Fw(2)p Fr(g)29 b Fw(and)h Fu(Q)3227 2989 y Fp(1)3295 2974 y Fw(=)d Fr(f)p Fu(a;)17 b(b)p Fr(g)p Fw(,)94 3090 y(where)34 b Fu(a)28 b Fw(:)g(1)f Fr(!)h Fw(2)k(and)h Fu(b)28 b Fw(:)g(2)f Fr(!)g Fw(1:)1671 3238 y(1)1829 3156 y Fq(a)1948 3185 y Fk(/)p Fj(/)p 1749 3186 200 4 v 1977 3238 a Fw(2)1833 3303 y Fq(b)1749 3228 y Fk(o)p Fj(o)p 1749 3229 V 2087 3231 a Fu(:)94 3442 y Fw(The)41 b(v)m(ertex)g(and)e(arro)m(w)h(spans)g(of)f Fu(Q)g Fw(are)g(giv)m(en)h(b)m(y)g Fu(R)g Fw(=)f Fu(K)2480 3406 y Fq(Q)2536 3415 y Fo(0)2614 3442 y Fw(=)g Fu(K)7 b(e)2864 3457 y Fp(1)2930 3442 y Fr(\010)27 b Fu(K)7 b(e)3169 3457 y Fp(2)3209 3442 y Fw(,)41 b(and)e Fu(A)h Fw(=)94 3558 y Fu(K)184 3522 y Fq(Q)240 3531 y Fo(1)307 3558 y Fw(=)27 b Fu(K)7 b(a)22 b Fr(\010)h Fu(K)7 b(b)p Fw(.)44 b(The)33 b(paths)g(in)f Fu(R)q Fr(hh)p Fu(A)p Fr(ii)f Fw(are)i Fu(e)1998 3573 y Fp(1)2037 3558 y Fw(,)g Fu(e)2142 3573 y Fp(2)2214 3558 y Fw(and)g(all)d(pro)s(ducts)j(of)f (the)h(generators)94 3675 y Fu(a)f Fw(and)f Fu(b)h Fw(in)f(whic)m(h)g (the)h(factors)f Fu(a)h Fw(and)f Fu(b)h Fw(alternate.)42 b(The)32 b(p)s(oten)m(tials)e(are)i(\(p)s(ossibly)e(in\014nite\))94 3791 y(linear)37 b(com)m(binations)f(of)h(the)i(elemen)m(ts)f(\()p Fu(ab)p Fw(\))1832 3755 y Fq(n)1917 3791 y Fw(and)g(\()p Fu(ba)p Fw(\))2280 3755 y Fq(n)2365 3791 y Fw(for)f(all)f Fu(n)h Fr(\025)g Fw(1.)59 b(Using)38 b(\()3322 3791 y SDict begin H.S end 3322 3791 a Fw(3.1)3446 3727 y SDict begin H.R end 3446 3727 a 3446 3791 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.3.1) cvn H.B /ANN pdfmark end 3446 3791 a Fw(\),)h(w)m(e)94 3907 y(obtain)153 4064 y Fu(@)204 4079 y Fq(a)246 4064 y Fw(\(\()p Fu(ab)p Fw(\))452 4023 y Fq(n)499 4064 y Fw(\))28 b(=)f Fu(@)719 4079 y Fq(a)762 4064 y Fw(\(\()p Fu(ba)p Fw(\))968 4023 y Fq(n)1015 4064 y Fw(\))h(=)f Fu(nb)p Fw(\()p Fu(ab)p Fw(\))1451 4023 y Fq(n)p Fn(\000)p Fp(1)1590 4064 y Fu(;)114 b(@)1782 4079 y Fq(b)1817 4064 y Fw(\(\()p Fu(ab)p Fw(\))2023 4023 y Fq(n)2070 4064 y Fw(\))28 b(=)f Fu(@)2290 4079 y Fq(b)2325 4064 y Fw(\(\()p Fu(ba)p Fw(\))2531 4023 y Fq(n)2579 4064 y Fw(\))g(=)h Fu(na)p Fw(\()p Fu(ba)p Fw(\))3025 4023 y Fq(n)p Fn(\000)p Fp(1)3260 4064 y Fw(\()p Fu(n)g Fr(\025)g Fw(1\))p Fu(:)94 4221 y Fw(Up)33 b(to)g(cyclical)e (equiv)-5 b(alence,)33 b(ev)m(ery)h(p)s(oten)m(tial)d(can)i(b)s(e)g (written)f(in)g(the)h(form)1411 4328 y Fn(1)1375 4358 y Fm(X)1380 4567 y Fq(n)p Fp(=1)1535 4452 y Fu(\013)1597 4467 y Fq(n)1644 4452 y Fw(\()p Fu(ab)p Fw(\))1812 4411 y Fq(n)1957 4452 y Fw(\()p Fu(\013)2057 4467 y Fq(n)2132 4452 y Fr(2)28 b Fu(K)7 b Fw(\))p Fu(:)194 4753 y Fw(Returning)24 b(to)g(the)h(general)f(theory)-8 b(,)27 b(it)c(is)h(clear)g(that)g(ev)m (ery)j(homomorphism)21 b(of)j Fu(R)q Fw(-algebras)1412 4910 y Fu(')k Fw(:)g Fu(R)q Fr(hh)p Fu(A)p Fr(ii)f(!)g Fu(R)q Fr(hh)p Fu(A)2243 4869 y Fn(0)2266 4910 y Fr(ii)94 5067 y Fw(sends)35 b(p)s(oten)m(tials)c(to)h(p)s(oten)m(tials.)94 5086 y SDict begin H.S end 94 5086 a 94 5086 a SDict begin 14 H.A end 94 5086 a 94 5086 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.3.7) cvn H.B /DEST pdfmark end 94 5086 a 155 x FA(Prop)s(osition)k(3.7.)42 b Fv(Every)35 b(isomorphism)e(of)i Fu(R)q Fv(-algebr)-5 b(as)1412 5398 y Fu(')28 b Fw(:)g Fu(R)q Fr(hh)p Fu(A)p Fr(ii)f(!)g Fu(R)q Fr(hh)p Fu(A)2243 5357 y Fn(0)2266 5398 y Fr(ii)94 5555 y Fv(sends)k Fu(J)9 b Fw(\()p Fu(S)d Fw(\))32 b Fv(onto)g Fu(J)9 b Fw(\()p Fu(')p Fw(\()p Fu(S)d Fw(\)\))p Fv(,)32 b(inducing)f(an)g(isomorphism)f (of)i(Jac)-5 b(obian)30 b(algebr)-5 b(as)31 b Fr(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\))28 b Fr(!)94 5671 y(P)8 b Fw(\()p Fu(A)282 5635 y Fn(0)306 5671 y Fu(;)17 b(')p Fw(\()p Fu(S)6 b Fw(\)\))p Fv(.)p eop end end %%Page: 11 11 TeXDict begin HPSdict begin 11 10 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.11) cvn H.B /DEST pdfmark end 94 218 a 1179 w Fs(QUIVERS)31 b(WITH)i(POTENTIALS)g(I)1102 b(11)94 417 y Fv(Pr)-5 b(o)g(of.)42 b Fw(W)-8 b(e)30 b(start)g(b)m(y)g(dev)m (eloping)g(some)g(\\di\013eren)m(tial)d(calculus")j(for)f(cyclic)g (deriv)-5 b(ativ)m(es.)43 b(W)-8 b(e)94 533 y(need)34 b(a)e(few)i(pieces)f(of)f(notation.)42 b(W)-8 b(e)33 b(set)1170 723 y Fu(R)q Fr(hh)p Fu(A)p Fr(ii)1485 707 y Fm(b)1474 723 y Fr(\012)p Fu(R)q Fr(hh)p Fu(A)p Fr(ii)27 b Fw(=)2011 628 y Fm(Y)1985 840 y Fq(d;e)p Fn(\025)p Fp(0)2164 723 y Fw(\()p Fu(A)2275 682 y Fq(d)2338 723 y Fr(\012)c Fu(A)2511 682 y Fq(e)2548 723 y Fw(\))94 1003 y(\(the)36 b(tensor)f(pro)s(duct)g(on)g(the)g(righ)m(t)f(is)g(o)m (v)m(er)i(the)f(base)h(\014eld)e Fu(K)7 b Fw(\),)35 b(and)g(view)g (this)g(space)h(as)f(a)94 1120 y(top)s(ological)c(v)m(ector)36 b(space)g(with)e(a)g(basic)g(system)i(of)e(op)s(en)h(neigh)m(b)s(orho)s (o)s(ds)e(of)h(0)h(formed)e(b)m(y)94 1236 y(the)24 b(sets)433 1161 y Fm(Q)527 1265 y Fq(d)p Fp(+)p Fq(e)p Fn(\025)p Fq(n)753 1236 y Fw(\()p Fu(A)864 1200 y Fq(d)907 1236 y Fr(\012)r Fu(A)1059 1200 y Fq(e)1096 1236 y Fw(\))e(for)g(all)f Fu(n)28 b Fr(\025)g Fw(0;)e(th)m(us,)f Fu(R)q Fr(h)p Fu(A)p Fr(i)r(\012)r Fu(R)q Fr(h)p Fu(A)p Fr(i)d Fw(is)h(dense)h(in)d Fu(R)q Fr(hh)p Fu(A)p Fr(ii)3265 1221 y Fm(b)3254 1236 y Fr(\012)p Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fw(.)94 1354 y(No)m(w,)34 b(for)e(ev)m(ery)i Fu(\030)e Fr(2)c Fu(A)992 1317 y Fq(?)1032 1354 y Fw(,)33 b(w)m(e)g(de\014ne)h(a)e(con)m(tin)m (uous)i Fu(K)7 b Fw(-linear)30 b(map)1206 1527 y(\001)1287 1542 y Fq(\030)1353 1527 y Fw(:)e Fu(R)q Fr(hh)p Fu(A)p Fr(ii)f(!)g Fu(R)q Fr(hh)p Fu(A)p Fr(ii)2181 1512 y Fm(b)2170 1527 y Fr(\012)p Fu(R)q Fr(hh)p Fu(A)p Fr(ii)94 1700 y Fw(b)m(y)34 b(setting)e(\001)631 1715 y Fq(\030)669 1700 y Fw(\()p Fu(e)p Fw(\))c(=)g(0)k(for)g Fu(e)c Fr(2)g Fu(R)h Fw(=)e Fu(A)1598 1664 y Fp(0)1638 1700 y Fw(,)32 b(and)94 1974 y(\(3.2\))875 1974 y SDict begin H.S end 875 1974 a 875 1974 a SDict begin 14 H.A end 875 1974 a 875 1974 a SDict begin [ /View [/XYZ H.V] /Dest (equation.3.2) cvn H.B /DEST pdfmark end 875 1974 a Fw(\001)956 1989 y Fq(\030)994 1974 y Fw(\()p Fu(a)1083 1989 y Fp(1)1139 1974 y Fr(\001)17 b(\001)g(\001)e Fu(a)1323 1989 y Fq(d)1363 1974 y Fw(\))28 b(=)1586 1850 y Fq(d)1533 1880 y Fm(X)1540 2092 y Fq(k)r Fp(=1)1693 1974 y Fu(\030)5 b Fw(\()p Fu(a)1830 1989 y Fq(k)1872 1974 y Fw(\))p Fu(a)1961 1989 y Fp(1)2017 1974 y Fr(\001)17 b(\001)g(\001)e Fu(a)2201 1989 y Fq(k)r Fn(\000)p Fp(1)2356 1974 y Fr(\012)23 b Fu(a)2507 1989 y Fq(k)r Fp(+1)2657 1974 y Fr(\001)17 b(\001)g(\001)e Fu(a)2841 1989 y Fq(d)94 2236 y Fw(for)28 b(an)m(y)g(path)g Fu(a)692 2251 y Fp(1)748 2236 y Fr(\001)17 b(\001)g(\001)e Fu(a)932 2251 y Fq(d)1001 2236 y Fw(of)27 b(length)g Fu(d)h Fr(\025)g Fw(1.)41 b(Note)28 b(that)g(\001)2220 2251 y Fq(\030)2286 2236 y Fw(do)s(es)g(not)g(dep)s(end)h(on)e(the)i(c) m(hoice)f(of)94 2353 y(a)34 b(path)h(basis.)48 b(W)-8 b(e)35 b(will)c(use)36 b(the)e(same)g(con)m(v)m(en)m(tion)i(as)e(for)g (cyclic)g(deriv)-5 b(ativ)m(es:)46 b(for)34 b Fu(a)d Fr(2)g Fu(Q)3595 2368 y Fp(1)3634 2353 y Fw(,)94 2469 y(w)m(e)j(write)e(\001)568 2484 y Fq(a)643 2469 y Fw(instead)g(of)g (\001)1171 2484 y Fq(a)1208 2465 y Fg(?)1249 2469 y Fw(.)44 b(F)-8 b(or)32 b(instance,)h(in)e(the)i(situation)e(of)i(Example)3107 2469 y SDict begin H.S end 3107 2469 a Fw(3.6)3231 2405 y SDict begin H.R end 3231 2405 a 3231 2469 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.3.6) cvn H.B /ANN pdfmark end 3231 2469 a Fw(,)g(w)m(e)g(ha)m(v)m(e)330 2717 y(\001)411 2732 y Fq(a)452 2717 y Fw(\(\()p Fu(ab)p Fw(\))658 2676 y Fq(n)706 2717 y Fw(\))28 b(=)926 2592 y Fq(n)875 2622 y Fm(X)883 2834 y Fq(k)r Fp(=1)1019 2717 y Fw(\()p Fu(ab)p Fw(\))1187 2676 y Fq(k)r Fn(\000)p Fp(1)1343 2717 y Fr(\012)22 b Fu(b)p Fw(\()p Fu(ab)p Fw(\))1651 2676 y Fq(n)p Fn(\000)p Fq(k)1793 2717 y Fu(;)114 b Fw(\001)2015 2732 y Fq(b)2049 2717 y Fw(\(\()p Fu(ab)p Fw(\))2255 2676 y Fq(n)2303 2717 y Fw(\))28 b(=)2523 2592 y Fq(n)2472 2622 y Fm(X)2480 2834 y Fq(k)r Fp(=1)2616 2717 y Fw(\()p Fu(ab)p Fw(\))2784 2676 y Fq(k)r Fn(\000)p Fp(1)2917 2717 y Fu(a)23 b Fr(\012)f Fw(\()p Fu(ab)p Fw(\))3258 2676 y Fq(n)p Fn(\000)p Fq(k)3399 2717 y Fu(:)194 2997 y Fw(Next,)k(w)m(e)d(denote)g(b)m(y)h(\()p Fu(f)5 b(;)17 b(g)t Fw(\))27 b Fr(7!)g Fu(f)11 b Fh(\003)p Fu(g)26 b Fw(a)c(con)m(tin)m(uous)h Fu(K)7 b Fw(-bilinear)20 b(map)i(\()p Fu(R)q Fr(hh)p Fu(A)p Fr(ii)3176 2982 y Fm(b)3165 2997 y Fr(\012)o Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fw(\))r Fr(\002)94 3113 y Fu(R)q Fr(hh)p Fu(A)p Fr(ii)27 b(!)h Fu(R)q Fr(hh)p Fu(A)p Fr(ii)j Fw(giv)m(en)i(b)m(y)94 3287 y(\(3.3\))1518 3287 y SDict begin H.S end 1518 3287 a 1518 3287 a SDict begin 14 H.A end 1518 3287 a 1518 3287 a SDict begin [ /View [/XYZ H.V] /Dest (equation.3.3) cvn H.B /DEST pdfmark end 1518 3287 a Fw(\()p Fu(u)22 b Fr(\012)g Fu(v)t Fw(\))p Fh(\003)p Fu(g)32 b Fw(=)27 b Fu(v)t(g)t(u)94 3460 y Fw(for)33 b Fu(u;)17 b(v)30 b Fr(2)e Fu(R)q Fr(h)p Fu(A)p Fr(i)p Fw(.)43 b(W)-8 b(e)33 b(are)g(no)m(w)g(ready)g(to)g (state)g(the)g(Leibniz)e(rule.)94 3485 y SDict begin H.S end 94 3485 a 94 3485 a SDict begin 14 H.A end 94 3485 a 94 3485 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.3.8) cvn H.B /DEST pdfmark end 94 3485 a 159 x FA(Lemma)47 b(3.8)f Fw(\(Cyclic)40 b(Leibniz)g(rule\))p FA(.)45 b Fv(L)-5 b(et)43 b Fu(f)52 b Fr(2)41 b Fu(R)q Fr(hh)p Fu(A)p Fr(ii)2346 3659 y Fq(i;j)2467 3644 y Fv(and)h Fu(g)i Fr(2)e Fu(R)q Fr(hh)p Fu(A)p Fr(ii)3167 3659 y Fq(j;i)3284 3644 y Fv(for)g(some)94 3760 y(vertic)-5 b(es)35 b Fu(i)g Fv(and)f Fu(j)6 b Fv(.)45 b(Then)34 b(for)h(every)g Fu(\030)d Fr(2)c Fu(A)1735 3724 y Fq(?)1774 3760 y Fv(,)35 b(we)g(have)94 3934 y Fw(\(3.4\))1223 3934 y SDict begin H.S end 1223 3934 a 1223 3934 a SDict begin 14 H.A end 1223 3934 a 1223 3934 a SDict begin [ /View [/XYZ H.V] /Dest (equation.3.4) cvn H.B /DEST pdfmark end 1223 3934 a Fu(@)1274 3949 y Fq(\030)1312 3934 y Fw(\()p Fu(f)11 b(g)t Fw(\))27 b(=)g(\001)1709 3949 y Fq(\030)1747 3934 y Fw(\()p Fu(f)11 b Fw(\))p Fh(\003)p Fu(g)26 b Fw(+)c(\001)2211 3949 y Fq(\030)2249 3934 y Fw(\()p Fu(g)t Fw(\))p Fh(\003)p Fu(f)5 b(:)94 4107 y Fv(Mor)-5 b(e)43 b(gener)-5 b(al)5 b(ly,)43 b(for)e(any)h(\014nite)g(se)-5 b(quenc)g(e)41 b(of)h(vertic)-5 b(es)41 b Fu(i)2338 4122 y Fp(1)2378 4107 y Fu(;)17 b(:)g(:)g(:)f(;)h(i)2630 4122 y Fq(d)2670 4107 y Fu(;)g(i)2747 4122 y Fq(d)p Fp(+1)2919 4107 y Fw(=)41 b Fu(i)3069 4122 y Fp(1)3150 4107 y Fv(and)h(for)f(any) 94 4223 y Fu(f)142 4238 y Fp(1)182 4223 y Fu(;)17 b(:)g(:)g(:)f(f)405 4238 y Fq(d)480 4223 y Fv(such)35 b(that)g Fu(f)949 4238 y Fq(k)1020 4223 y Fr(2)28 b Fu(R)q Fr(hh)p Fu(A)p Fr(ii)1418 4238 y Fq(i)1442 4250 y Fg(k)1479 4238 y Fq(;i)1523 4250 y Fg(k)q Fo(+1)1642 4223 y Fv(,)35 b(we)f(have)94 4503 y Fw(\(3.5\))835 4503 y SDict begin H.S end 835 4503 a 835 4503 a SDict begin 14 H.A end 835 4503 a 835 4503 a SDict begin [ /View [/XYZ H.V] /Dest (equation.3.5) cvn H.B /DEST pdfmark end 835 4503 a Fu(@)886 4518 y Fq(\030)925 4503 y Fw(\()p Fu(f)1011 4518 y Fp(1)1067 4503 y Fr(\001)17 b(\001)g(\001)d Fu(f)1247 4518 y Fq(d)1288 4503 y Fw(\))28 b(=)1511 4378 y Fq(d)1457 4408 y Fm(X)1465 4620 y Fq(k)r Fp(=1)1618 4503 y Fw(\001)1699 4518 y Fq(\030)1737 4503 y Fw(\()p Fu(f)1823 4518 y Fq(k)1865 4503 y Fw(\))p Fh(\003)p Fw(\()p Fu(f)2066 4518 y Fq(k)r Fp(+1)2216 4503 y Fr(\001)17 b(\001)g(\001)e Fu(f)2397 4518 y Fq(d)2438 4503 y Fu(f)2486 4518 y Fp(1)2542 4503 y Fr(\001)i(\001)g(\001)e Fu(f)2723 4518 y Fq(k)r Fn(\000)p Fp(1)2856 4503 y Fw(\))p Fu(:)94 4781 y Fv(Pr)-5 b(o)g(of.)42 b Fw(It)25 b(is)f(enough)h(to)g(c)m(hec)m (k)i(\()1314 4781 y SDict begin H.S end 1314 4781 a Fw(3.4)1439 4717 y SDict begin H.R end 1439 4717 a 1439 4781 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.3.4) cvn H.B /ANN pdfmark end 1439 4781 a Fw(\))d(in)h(the)g(case)h(where)g Fu(f)38 b Fw(=)28 b Fu(a)2482 4796 y Fp(1)2538 4781 y Fr(\001)17 b(\001)g(\001)e Fu(a)2722 4796 y Fq(d)2787 4781 y Fw(and)25 b Fu(g)31 b Fw(=)d Fu(a)3202 4796 y Fq(d)p Fp(+1)3349 4781 y Fr(\001)17 b(\001)g(\001)e Fu(a)3533 4796 y Fq(d)p Fp(+)p Fq(s)94 4897 y Fw(are)32 b(t)m(w)m(o)g(paths)g (suc)m(h)h(that)e Fu(t)p Fw(\()p Fu(a)1257 4912 y Fq(d)1298 4897 y Fw(\))d(=)f Fu(h)p Fw(\()p Fu(a)1612 4912 y Fq(d)p Fp(+1)1743 4897 y Fw(\))32 b(and)f Fu(t)p Fw(\()p Fu(a)2125 4912 y Fq(d)p Fp(+)p Fq(s)2254 4897 y Fw(\))c(=)h Fu(h)p Fw(\()p Fu(a)2568 4912 y Fp(1)2608 4897 y Fw(\).)43 b(Using)31 b(\()3027 4897 y SDict begin H.S end 3027 4897 a Fw(3.1)3151 4833 y SDict begin H.R end 3151 4833 a 3151 4897 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.3.1) cvn H.B /ANN pdfmark end 3151 4897 a Fw(\),)h(w)m(e)g(obtain)1004 5168 y Fu(@)1055 5183 y Fq(\030)1094 5168 y Fw(\()p Fu(f)11 b(g)t Fw(\))26 b(=)1420 5044 y Fq(d)p Fp(+)p Fq(s)1410 5074 y Fm(X)1417 5286 y Fq(k)r Fp(=1)1570 5168 y Fu(\030)5 b Fw(\()p Fu(a)1707 5183 y Fq(k)1750 5168 y Fw(\))p Fu(a)1839 5183 y Fq(k)r Fp(+1)1988 5168 y Fr(\001)17 b(\001)g(\001)e Fu(a)2172 5183 y Fq(d)p Fp(+)p Fq(s)2300 5168 y Fu(a)2351 5183 y Fp(1)2407 5168 y Fr(\001)i(\001)g(\001)e Fu(a)2591 5183 y Fq(k)r Fn(\000)p Fp(1)2724 5168 y Fu(:)94 5436 y Fw(Comparing)29 b(this)h(expression)i(with)e(\()1510 5436 y SDict begin H.S end 1510 5436 a Fw(3.2)1635 5372 y SDict begin H.R end 1635 5372 a 1635 5436 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.3.2) cvn H.B /ANN pdfmark end 1635 5436 a Fw(\))g(and)g(\()1928 5436 y SDict begin H.S end 1928 5436 a Fw(3.3)2053 5372 y SDict begin H.R end 2053 5372 a 2053 5436 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.3.3) cvn H.B /ANN pdfmark end 2053 5436 a Fw(\),)h(w)m(e)g(see)h(that)e(the)h(part)f(of)g(the)h(last)e (sum)94 5552 y(where)36 b Fu(k)h Fw(runs)e(from)e(1)h(to)g Fu(d)g Fw(\(resp.)49 b(from)33 b Fu(d)23 b Fw(+)g(1)34 b(to)f Fu(d)23 b Fw(+)g Fu(s)p Fw(\))34 b(is)g(equal)g(to)g(\001)2953 5567 y Fq(\030)2991 5552 y Fw(\()p Fu(f)11 b Fw(\))p Fh(\003)p Fu(g)38 b Fw(\(resp.)49 b(to)94 5668 y(\001)175 5683 y Fq(\030)214 5668 y Fw(\()p Fu(g)t Fw(\))p Fh(\003)p Fu(f)11 b Fw(\),)26 b(pro)m(ving)g(\()952 5668 y SDict begin H.S end 952 5668 a Fw(3.4)1077 5604 y SDict begin H.R end 1077 5604 a 1077 5668 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.3.4) cvn H.B /ANN pdfmark end 1077 5668 a Fw(\).)41 b(The)27 b(iden)m(tit)m(y)g(\()1770 5668 y SDict begin H.S end 1770 5668 a Fw(3.5)1894 5604 y SDict begin H.R end 1894 5604 a 1894 5668 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.3.5) cvn H.B /ANN pdfmark end 1894 5668 a Fw(\))f(follo)m(ws)f(from)g(\()2534 5668 y SDict begin H.S end 2534 5668 a Fw(3.4)2659 5604 y SDict begin H.R end 2659 5604 a 2659 5668 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.3.4) cvn H.B /ANN pdfmark end 2659 5668 a Fw(\))h(b)m(y)h(induction)e(on)i Fu(d)p Fw(.)97 b Fh(\003)94 5766 y SDict begin H.S end 94 5766 a 94 5766 a SDict begin 14 H.A end 94 5766 a 94 5766 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.3.9) cvn H.B /DEST pdfmark end 94 5766 a eop end end %%Page: 12 12 TeXDict begin HPSdict begin 12 11 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.12) cvn H.B /DEST pdfmark end 94 218 a Fs(12)492 b(HARM)25 b(DERKSEN,)f(JERZY)i(WEYMAN,)f(AND)f(ANDREI)g(ZELEVINSKY)94 417 y FA(Lemma)33 b(3.9)h Fw(\(Cyclic)29 b(c)m(hain)f(rule\))p FA(.)39 b Fv(Supp)-5 b(ose)31 b(that)h Fu(')27 b Fw(:)h Fu(R)q Fr(hh)p Fu(A)p Fr(ii)f(!)g Fu(R)q Fr(hh)p Fu(A)2912 381 y Fn(0)2935 417 y Fr(ii)k Fv(is)g(a)h(homomor-)94 533 y(phism)i(of)g Fu(R)q Fv(-algebr)-5 b(as)33 b(as)h(in)g(Pr)-5 b(op)g(osition)1731 533 y SDict begin H.S end 1731 533 a Fv(2.5)1861 469 y SDict begin H.R end 1861 469 a 1861 533 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.2.5) cvn H.B /ANN pdfmark end 1861 533 a Fv(.)44 b(Then,)34 b(for)g(every)g(p)-5 b(otential)34 b Fu(S)f Fr(2)28 b Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fw(\))3558 548 y Fp(cyc)94 649 y Fv(and)35 b Fu(\030)d Fr(2)c Fu(A)526 613 y Fn(0)p Fq(?)585 649 y Fv(,)34 b(we)h(have:)94 836 y Fw(\(3.6\))1099 836 y SDict begin H.S end 1099 836 a 1099 836 a SDict begin 14 H.A end 1099 836 a 1099 836 a SDict begin [ /View [/XYZ H.V] /Dest (equation.3.6) cvn H.B /DEST pdfmark end 1099 836 a Fu(@)1150 851 y Fq(\030)1188 836 y Fw(\()p Fu(')p Fw(\()p Fu(S)6 b Fw(\)\))27 b(=)1616 741 y Fm(X)1601 953 y Fq(a)p Fn(2)p Fq(Q)1741 962 y Fo(1)1792 836 y Fw(\001)1873 851 y Fq(\030)1911 836 y Fw(\()p Fu(')p Fw(\()p Fu(a)p Fw(\)\))p Fh(\003)p Fu(')p Fw(\()p Fu(@)2408 851 y Fq(a)2450 836 y Fw(\()p Fu(S)6 b Fw(\)\))p Fu(:)94 1119 y Fv(Pr)-5 b(o)g(of.)42 b Fw(It)f(su\016ces)j(to)d(treat)g(the)h(case)g(where)g Fu(S)49 b Fw(=)42 b Fu(a)2183 1134 y Fp(1)2240 1119 y Fr(\001)17 b(\001)g(\001)d Fu(a)2423 1134 y Fq(d)2505 1119 y Fw(is)41 b(a)g(cyclic)g(path.)70 b(Applying)94 1235 y(\()132 1235 y SDict begin H.S end 132 1235 a Fw(3.5)257 1171 y SDict begin H.R end 257 1171 a 257 1235 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.3.5) cvn H.B /ANN pdfmark end 257 1235 a Fw(\))33 b(and)f(\()555 1235 y SDict begin H.S end 555 1235 a Fw(3.1)680 1171 y SDict begin H.R end 680 1171 a 680 1235 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.3.1) cvn H.B /ANN pdfmark end 680 1235 a Fw(\),)g(w)m(e)i(obtain)684 1503 y Fu(@)735 1518 y Fq(\030)773 1503 y Fw(\()p Fu(')p Fw(\()p Fu(S)6 b Fw(\)\))27 b(=)1240 1379 y Fq(d)1186 1408 y Fm(X)1193 1621 y Fq(k)r Fp(=1)1346 1503 y Fw(\001)1427 1518 y Fq(\030)1465 1503 y Fw(\()p Fu(')p Fw(\()p Fu(a)1656 1518 y Fq(k)1699 1503 y Fw(\)\))p Fh(\003)p Fw(\()p Fu(')p Fw(\()p Fu(a)2043 1518 y Fq(k)r Fp(+1)2193 1503 y Fr(\001)17 b(\001)g(\001)e Fu(a)2377 1518 y Fq(d)2417 1503 y Fu(a)2468 1518 y Fp(1)2524 1503 y Fr(\001)i(\001)g(\001)e Fu(a)2708 1518 y Fq(k)r Fn(\000)p Fp(1)2841 1503 y Fw(\)\))1082 1772 y(=)1201 1678 y Fm(X)1186 1889 y Fq(a)p Fn(2)p Fq(Q)1326 1898 y Fo(1)1377 1772 y Fw(\001)1458 1787 y Fq(\030)1496 1772 y Fw(\()p Fu(')p Fw(\()p Fu(a)p Fw(\)\))p Fh(\003)p Fu(')p Fw(\()1983 1678 y Fm(X)1942 1890 y Fq(k)r Fp(:)p Fq(a)2038 1902 y Fg(k)2076 1890 y Fp(=)p Fq(a)2185 1772 y Fu(a)2236 1787 y Fq(k)r Fp(+1)2386 1772 y Fr(\001)i(\001)g(\001)d Fu(a)2569 1787 y Fq(d)2610 1772 y Fu(a)2661 1787 y Fp(1)2717 1772 y Fr(\001)j(\001)g(\001)e Fu(a)2901 1787 y Fq(k)r Fn(\000)p Fp(1)3034 1772 y Fw(\))1082 2054 y(=)1201 1959 y Fm(X)1186 2171 y Fq(a)p Fn(2)p Fq(Q)1326 2180 y Fo(1)1377 2054 y Fw(\001)1458 2069 y Fq(\030)1496 2054 y Fw(\()p Fu(')p Fw(\()p Fu(a)p Fw(\)\))p Fh(\003)p Fu(')p Fw(\()p Fu(@)1993 2069 y Fq(a)2035 2054 y Fw(\()p Fu(S)6 b Fw(\)\))p Fu(;)94 2325 y Fw(as)33 b(desired.)3045 b Fh(\003)194 2508 y Fw(No)m(w)32 b(w)m(e)g(are)f(ready)h(to)e(pro)m(v)m(e)j(Prop)s (osition)1887 2508 y SDict begin H.S end 1887 2508 a Fw(3.7)2012 2444 y SDict begin H.R end 2012 2444 a 2012 2508 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.3.7) cvn H.B /ANN pdfmark end 2012 2508 a Fw(.)43 b(By)31 b(Lemma)2580 2508 y SDict begin H.S end 2580 2508 a Fw(3.9)2705 2444 y SDict begin H.R end 2705 2444 a 2705 2508 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.3.9) cvn H.B /ANN pdfmark end 2705 2508 a Fw(,)g(for)g(ev)m(ery)i Fu(\030)f Fr(2)c Fu(A)3409 2472 y Fn(0)p Fq(?)3468 2508 y Fw(,)j(the)94 2624 y(elemen)m(t)39 b Fu(@)512 2639 y Fq(\030)551 2624 y Fw(\()p Fu(')p Fw(\()p Fu(S)6 b Fw(\)\))38 b(lies)g(in)g(the)i(ideal) e(generated)h(b)m(y)h(the)g(elemen)m(ts)f Fu(')p Fw(\()p Fu(@)2906 2639 y Fq(a)2948 2624 y Fw(\()p Fu(S)6 b Fw(\)\))39 b(for)f Fu(a)h Fr(2)h Fu(Q)3595 2639 y Fp(1)3634 2624 y Fw(,)94 2740 y(hence,)35 b(it)c(lies)h(in)f Fu(')p Fw(\()p Fu(J)9 b Fw(\()p Fu(S)d Fw(\)\).)43 b(Th)m(us,)35 b(w)m(e)e(ha)m(v)m(e)h(the)f(inclusion)1453 2911 y Fu(J)9 b Fw(\()p Fu(')p Fw(\()p Fu(S)d Fw(\)\))28 b Fr(\022)g Fu(')p Fw(\()p Fu(J)9 b Fw(\()p Fu(S)d Fw(\)\))p Fu(:)94 3081 y Fw(W)-8 b(e)33 b(can)g(also)f(apply)g(this)g(to)h(the)g(in)m(v)m (erse)h(isomorphism)c Fu(')2340 3045 y Fn(\000)p Fp(1)2467 3081 y Fw(and)i(the)h(p)s(oten)m(tial)e Fu(')p Fw(\()p Fu(S)6 b Fw(\):)1070 3252 y Fu(J)j Fw(\()p Fu(S)d Fw(\))28 b(=)f Fu(J)9 b Fw(\()p Fu(')1571 3211 y Fn(\000)p Fp(1)1666 3252 y Fw(\()p Fu(')p Fw(\()p Fu(S)d Fw(\)\)\))27 b Fr(\022)h Fu(')2182 3211 y Fn(\000)p Fp(1)2276 3252 y Fw(\()p Fu(J)9 b Fw(\()p Fu(')p Fw(\()p Fu(S)d Fw(\)\))p Fu(:)94 3423 y Fw(Applying)32 b Fu(')h Fw(to)f(b)s(oth)g(sides)h(yields)1453 3593 y Fu(')p Fw(\()p Fu(J)9 b Fw(\()p Fu(S)d Fw(\)\))28 b Fr(\022)g Fu(J)9 b Fw(\()p Fu(')p Fw(\()p Fu(S)d Fw(\)\))p Fu(;)94 3764 y Fw(completing)31 b(the)i(pro)s(of.)2573 b Fh(\003)94 3903 y SDict begin H.S end 94 3903 a 94 3903 a SDict begin 14 H.A end 94 3903 a 94 3903 a SDict begin [ /View [/XYZ H.V] /Dest (section.4) cvn H.B /DEST pdfmark end 94 3903 a 1192 4000 a Fw(4.)49 b Fx(Quivers)37 b(with)h(potentials)194 4174 y Fw(W)-8 b(e)33 b(no)m(w)g(in)m(tro)s (duce)g(our)f(main)f(ob)5 b(jects)34 b(of)e(study)-8 b(.)94 4193 y SDict begin H.S end 94 4193 a 94 4193 a SDict begin 14 H.A end 94 4193 a 94 4193 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.4.1) cvn H.B /DEST pdfmark end 94 4193 a 163 x FA(De\014nition)40 b(4.1.)k Fw(Supp)s(ose)36 b Fu(Q)g Fw(is)g(a)f(quiv)m(er)i(with)e(the)h(arro)m(w)g(span)g Fu(A)p Fw(,)h(and)f Fu(S)j Fr(2)33 b Fu(R)q Fr(hh)p Fu(A)p Fr(ii)3558 4371 y Fp(cyc)94 4472 y Fw(is)j(a)f(p)s(oten)m(tial.)51 b(W)-8 b(e)37 b(sa)m(y)f(that)g(a)f(pair)g(\()p Fu(Q;)17 b(S)6 b Fw(\))35 b(\(or)h(\()p Fu(A;)17 b(S)6 b Fw(\)\))35 b(is)g(a)g Fv(quiver)j(with)g(p)-5 b(otential)45 b Fw(\(QP)94 4588 y(for)33 b(short\))f(if)g(it)f(satis\014es)j(the)f(follo)m(wing)c (t)m(w)m(o)34 b(conditions:)94 4759 y(\(4.1\))718 4759 y SDict begin H.S end 718 4759 a 718 4759 a SDict begin 14 H.A end 718 4759 a 718 4759 a SDict begin [ /View [/XYZ H.V] /Dest (equation.4.1) cvn H.B /DEST pdfmark end 718 4759 a Fw(The)g(quiv)m(er)f Fu(Q)g Fw(has)g(no)f(lo)s(ops,)g(i.e.,)g Fu(A)2170 4774 y Fq(i;i)2270 4759 y Fw(=)27 b(0)32 b(for)h(all)d Fu(i)e Fr(2)g Fu(Q)2971 4774 y Fp(0)3011 4759 y Fw(.)94 4983 y(\(4.2\))373 4983 y SDict begin H.S end 373 4983 a 373 4983 a SDict begin 14 H.A end 373 4983 a 373 4983 a SDict begin [ /View [/XYZ H.V] /Dest (equation.4.2) cvn H.B /DEST pdfmark end 373 4983 a Fw(No)33 b(t)m(w)m(o)g(cyclically)e(equiv)-5 b(alen)m(t)32 b(cyclic)g(paths)h(app)s(ear)g(in)f(the)h(decomp)s (osition)d(of)i Fu(S)6 b Fw(.)194 5165 y(In)29 b(view)h(of)e(\()679 5165 y SDict begin H.S end 679 5165 a Fw(4.1)804 5101 y SDict begin H.R end 804 5101 a 804 5165 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.4.1) cvn H.B /ANN pdfmark end 804 5165 a Fw(\),)h(ev)m(ery)i(p)s(oten)m(tial)c Fu(S)35 b Fw(b)s(elongs)28 b(to)h Fl(m)p Fw(\()p Fu(A)p Fw(\))2342 5129 y Fp(2)2382 5165 y Fw(;)h(and)f(condition)e(\()3087 5165 y SDict begin H.S end 3087 5165 a Fw(4.2)3212 5101 y SDict begin H.R end 3212 5101 a 3212 5165 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.4.2) cvn H.B /ANN pdfmark end 3212 5165 a Fw(\))i(excludes,)94 5282 y(for)k(instance,)g(an)m (y)g(non-zero)f(p)s(oten)m(tial)f Fu(S)39 b Fw(cyclically)30 b(equiv)-5 b(alen)m(t)33 b(to)f(0.)94 5301 y SDict begin H.S end 94 5301 a 94 5301 a SDict begin 14 H.A end 94 5301 a 94 5301 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.4.2) cvn H.B /DEST pdfmark end 94 5301 a 163 x FA(De\014nition)24 b(4.2.)34 b Fw(Let)22 b(\()p Fu(A;)17 b(S)6 b Fw(\))22 b(and)g(\()p Fu(A)1548 5428 y Fn(0)1571 5464 y Fu(;)17 b(S)1681 5428 y Fn(0)1704 5464 y Fw(\))22 b(b)s(e)g(QPs)h(on)f(the)g (same)g(v)m(ertex)i(set)f Fu(Q)3109 5479 y Fp(0)3148 5464 y Fw(.)40 b(By)23 b(a)f Fv(right-)94 5580 y(e)-5 b(quivalenc)g(e)42 b Fw(b)s(et)m(w)m(een)37 b(\()p Fu(A;)17 b(S)6 b Fw(\))35 b(and)g(\()p Fu(A)1584 5544 y Fn(0)1608 5580 y Fu(;)17 b(S)1718 5544 y Fn(0)1741 5580 y Fw(\))35 b(w)m(e)h(mean)f(an)g(isomorphism)e(of)i Fu(R)q Fw(-algebras)f Fu(')e Fw(:)94 5696 y Fu(R)q Fr(hh)p Fu(A)p Fr(ii)27 b(!)h Fu(R)q Fr(hh)p Fu(A)779 5660 y Fn(0)802 5696 y Fr(ii)k Fw(suc)m(h)i(that)e Fu(')p Fw(\()p Fu(S)6 b Fw(\))32 b(is)g(cyclically)f(equiv)-5 b(alen)m(t)32 b(to)g Fu(S)2750 5660 y Fn(0)2806 5696 y Fw(\(see)i(De\014nition)3456 5696 y SDict begin H.S end 3456 5696 a Fw(3.2)3580 5632 y SDict begin H.R end 3580 5632 a 3580 5696 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.3.2) cvn H.B /ANN pdfmark end 3580 5696 a Fw(\).)p eop end end %%Page: 13 13 TeXDict begin HPSdict begin 13 12 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.13) cvn H.B /DEST pdfmark end 94 218 a 1179 w Fs(QUIVERS)31 b(WITH)i(POTENTIALS)g(I)1102 b(13)194 417 y Fw(In)36 b(view)h(of)e(Prop)s(osition)1191 417 y SDict begin H.S end 1191 417 a Fw(3.5)1315 353 y SDict begin H.R end 1315 353 a 1315 417 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.3.5) cvn H.B /ANN pdfmark end 1315 417 a Fw(,)i(an)m(y)g(homomorphism)c(of)j Fu(R)q Fw(-algebras)f Fu(R)q Fr(hh)p Fu(A)p Fr(ii)e(!)g Fu(R)q Fr(hh)p Fu(A)3561 381 y Fn(0)3584 417 y Fr(ii)94 533 y Fw(sends)43 b(cyclically)d(equiv)-5 b(alen)m(t)41 b(p)s(oten)m(tials) f(to)h(cyclically)e(equiv)-5 b(alen)m(t)41 b(ones.)70 b(It)42 b(follo)m(ws)e(that)94 649 y(righ)m(t-equiv)-5 b(alences)24 b(of)g(QPs)h(ha)m(v)m(e)h(the)f(exp)s(ected)h(prop)s (erties:)39 b(the)25 b(comp)s(osition)d(of)i(t)m(w)m(o)h(righ)m(t-)94 766 y(equiv)-5 b(alences,)27 b(as)e(w)m(ell)e(as)i(the)f(in)m(v)m(erse) i(of)e(a)g(righ)m(t-equiv)-5 b(alence,)25 b(is)e(again)g(a)h(righ)m (t-equiv)-5 b(alence.)94 882 y(Note)41 b(also)e(that)i(an)f (isomorphism)e Fu(')j Fw(:)g Fu(R)q Fr(hh)p Fu(A)p Fr(ii)f(!)h Fu(R)q Fr(hh)p Fu(A)2366 846 y Fn(0)2389 882 y Fr(ii)e Fw(induces)j(an)e(isomorphism)e(of)94 998 y Fu(R)q Fw(-bimo)s(dules)28 b Fu(A)i Fw(and)g Fu(A)1026 962 y Fn(0)1080 998 y Fw(\(cf.)42 b(Prop)s(osition)1783 998 y SDict begin H.S end 1783 998 a Fw(2.5)1907 934 y SDict begin H.R end 1907 934 a 1907 998 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.2.5) cvn H.B /ANN pdfmark end 1907 998 a Fw(\),)31 b(so)f(in)f(dealing)g (with)g(righ)m(t-equiv)-5 b(alen)m(t)29 b(QPs)94 1114 y(w)m(e)34 b(can)f(assume)g(without)f(loss)g(of)h(generalit)m(y)e(that) i Fu(A)28 b Fw(=)f Fu(A)2352 1078 y Fn(0)2376 1114 y Fw(.)194 1231 y(In)i(view)g(of)g(Prop)s(ositions)1200 1231 y SDict begin H.S end 1200 1231 a Fw(3.3)1325 1166 y SDict begin H.R end 1325 1166 a 1325 1231 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.3.3) cvn H.B /ANN pdfmark end 1325 1231 a 29 w Fw(and)1540 1231 y SDict begin H.S end 1540 1231 a Fw(3.7)1664 1166 y SDict begin H.R end 1664 1166 a 1664 1231 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.3.7) cvn H.B /ANN pdfmark end 1664 1231 a Fw(,)h(an)m(y)f(righ)m(t-equiv)-5 b(alence)28 b(of)h(QPs)g(\()p Fu(A;)17 b(S)6 b Fw(\))3251 1203 y Fr(\030)3252 1235 y Fw(=)3356 1231 y(\()p Fu(A)3467 1194 y Fn(0)3491 1231 y Fu(;)17 b(S)3601 1194 y Fn(0)3623 1231 y Fw(\))94 1347 y(induces)42 b(an)e(isomorphism)e(of)i(the)h(Jacobian)f(ideals)f Fu(J)9 b Fw(\()p Fu(S)d Fw(\))2407 1319 y Fr(\030)2408 1351 y Fw(=)2525 1347 y Fu(J)j Fw(\()p Fu(S)2692 1311 y Fn(0)2715 1347 y Fw(\))41 b(and)f(of)g(the)h(Jacobian)94 1463 y(algebras)32 b Fr(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\))838 1435 y Fr(\030)839 1467 y Fw(=)943 1463 y Fr(P)i Fw(\()p Fu(A)1131 1427 y Fn(0)1155 1463 y Fu(;)17 b(S)1265 1427 y Fn(0)1288 1463 y Fw(\).)194 1579 y(F)-8 b(or)39 b(ev)m(ery)j(t)m(w)m(o)f(QPs)f(\()p Fu(A;)17 b(S)6 b Fw(\))40 b(and)g(\()p Fu(A)1660 1543 y Fn(0)1683 1579 y Fu(;)17 b(S)1793 1543 y Fn(0)1816 1579 y Fw(\))40 b(\(on)g(the)g(same)g(set)h(of)e(v)m(ertices)j Fu(Q)3218 1594 y Fp(0)3257 1579 y Fw(\),)g(w)m(e)f(can)94 1696 y(form)f(their)g Fv(dir)-5 b(e)g(ct)42 b(sum)47 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\))27 b Fr(\010)h Fw(\()p Fu(A)1570 1659 y Fn(0)1594 1696 y Fu(;)17 b(S)1704 1659 y Fn(0)1727 1696 y Fw(\))41 b(=)g(\()p Fu(A)27 b Fr(\010)h Fu(A)2239 1659 y Fn(0)2263 1696 y Fu(;)17 b(S)33 b Fw(+)27 b Fu(S)2569 1659 y Fn(0)2592 1696 y Fw(\);)45 b(it)39 b(is)h(w)m(ell-de\014ned)h (since)94 1812 y(b)s(oth)g(complete)f(path)h(algebras)f Fu(R)q Fr(hh)p Fu(A)p Fr(ii)g Fw(and)h Fu(R)q Fr(hh)p Fu(A)2145 1776 y Fn(0)2168 1812 y Fr(ii)g Fw(ha)m(v)m(e)h(canonical)d (em)m(b)s(eddings)i(in)m(to)94 1928 y Fu(R)q Fr(hh)p Fu(A)22 b Fr(\010)h Fu(A)515 1892 y Fn(0)538 1928 y Fr(ii)32 b Fw(as)h(closed)g Fu(R)q Fw(-subalgebras.)194 2044 y(W)-8 b(e)44 b(start)f(our)g(analysis)f(of)h(QPs)g(with)g(the)h(case)g Fu(S)51 b Fr(2)46 b Fu(A)2448 2008 y Fp(2)2488 2044 y Fw(.)75 b(In)43 b(this)g(case,)j Fu(J)9 b Fw(\()p Fu(S)d Fw(\))43 b(is)g(the)94 2160 y(closure)33 b(of)f(the)h(ideal)e (generated)j(b)m(y)f(the)g(subspace)94 2331 y(\(4.3\))1287 2331 y SDict begin H.S end 1287 2331 a 1287 2331 a SDict begin 14 H.A end 1287 2331 a 1287 2331 a SDict begin [ /View [/XYZ H.V] /Dest (equation.4.3) cvn H.B /DEST pdfmark end 1287 2331 a Fu(@)5 b(S)34 b Fw(=)28 b Fr(f)p Fu(@)1642 2346 y Fq(\030)1680 2331 y Fw(\()p Fu(S)6 b Fw(\))27 b Fr(j)g Fu(\030)33 b Fr(2)28 b Fu(A)2147 2290 y Fq(?)2186 2331 y Fr(g)g(\022)g Fu(A:)94 2412 y SDict begin H.S end 94 2412 a 94 2412 a SDict begin 14 H.A end 94 2412 a 94 2412 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.4.3) cvn H.B /DEST pdfmark end 94 2412 a 101 x FA(De\014nition)48 b(4.3.)e Fw(W)-8 b(e)43 b(sa)m(y)g(that)f(a)g(QP)g(\()p Fu(A;)17 b(S)6 b Fw(\))41 b(is)h Fv(trivial)52 b Fw(if)41 b Fu(S)50 b Fr(2)44 b Fu(A)2806 2476 y Fp(2)2846 2513 y Fw(,)g(and)f Fu(@)5 b(S)50 b Fw(=)44 b Fu(A)p Fw(,)h(or,)94 2629 y(equiv)-5 b(alen)m(tly)d(,)32 b Fr(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\))28 b(=)g Fu(R)q Fw(.)194 2811 y(The)34 b(follo)m(wing)c(description)i(of)g(trivial)e(QPs)j(is)f(seen) i(b)m(y)f(standard)g(linear)e(algebra.)94 2830 y SDict begin H.S end 94 2830 a 94 2830 a SDict begin 14 H.A end 94 2830 a 94 2830 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.4.4) cvn H.B /DEST pdfmark end 94 2830 a 163 x FA(Prop)s(osition)g (4.4.)38 b Fv(A)31 b(QP)f Fw(\()p Fu(A;)17 b(S)6 b Fw(\))30 b Fv(with)g Fu(S)k Fr(2)28 b Fu(A)1938 2957 y Fp(2)2008 2993 y Fv(is)i(trivial)h(if)f(and)g(only)g(if)h(the)f(set)h(of)f(arr)-5 b(ows)94 3109 y Fu(Q)171 3124 y Fp(1)253 3109 y Fv(c)g(onsists)40 b(of)h Fw(2)p Fu(N)52 b Fv(distinct)41 b(arr)-5 b(ows)41 b Fu(a)1649 3124 y Fp(1)1688 3109 y Fu(;)17 b(b)1773 3124 y Fp(1)1813 3109 y Fu(;)g(:)g(:)g(:)f(;)h(a)2083 3124 y Fq(N)2150 3109 y Fu(;)g(b)2235 3124 y Fq(N)2344 3109 y Fv(such)41 b(that)h(e)-5 b(ach)40 b Fu(a)3054 3124 y Fq(k)3097 3109 y Fu(b)3138 3124 y Fq(k)3223 3109 y Fv(is)h(a)g(cyclic)94 3225 y Fw(2)p Fv(-p)-5 b(ath,)49 b(and)c(ther)-5 b(e)47 b(is)f(a)g(change)f(of)h(arr)-5 b(ows)45 b Fu(')h Fv(\(se)-5 b(e)46 b(De\014nition)2681 3225 y SDict begin H.S end 2681 3225 a Fv(2.6)2810 3161 y SDict begin H.R end 2810 3161 a 2810 3225 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.2.6) cvn H.B /ANN pdfmark end 2810 3225 a Fv(\))g(such)g(that)h Fu(')p Fw(\()p Fu(S)6 b Fw(\))46 b Fv(is)94 3341 y(cyclic)-5 b(al)5 b(ly)35 b(e)-5 b(quivalent)34 b(to)h Fu(a)1134 3356 y Fp(1)1174 3341 y Fu(b)1215 3356 y Fp(1)1277 3341 y Fw(+)22 b Fr(\001)17 b(\001)g(\001)j Fw(+)i Fu(a)1662 3356 y Fq(N)1730 3341 y Fu(b)1771 3356 y Fq(N)1839 3341 y Fv(.)194 3523 y Fw(Returning)31 b(to)h(general)f(QPs,)i(w)m(e)g(no)m(w)f(sho)m (w)h(that)f(taking)f(direct)h(sums)g(with)f(trivial)f(ones)94 3640 y(do)s(es)k(not)e(a\013ect)h(the)g(Jacobian)f(algebra.)94 3659 y SDict begin H.S end 94 3659 a 94 3659 a SDict begin 14 H.A end 94 3659 a 94 3659 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.4.5) cvn H.B /DEST pdfmark end 94 3659 a 163 x FA(Prop)s(osition)48 b(4.5.)f Fv(If)d Fw(\()p Fu(A;)17 b(S)6 b Fw(\))44 b Fv(is)g(an)g(arbitr)-5 b(ary)45 b(QP,)f(and)g Fw(\()p Fu(C)r(;)17 b(T)d Fw(\))43 b Fv(is)h(a)g(trivial) g(one,)j(then)94 3938 y(the)33 b(c)-5 b(anonic)g(al)31 b(emb)-5 b(e)g(dding)31 b Fu(R)q Fr(hh)p Fu(A)p Fr(ii)26 b(!)i Fu(R)q Fr(hh)p Fu(A)16 b Fr(\010)h Fu(C)7 b Fr(ii)32 b Fv(induc)-5 b(es)32 b(an)g(isomorphism)e(of)i(Jac)-5 b(obian)94 4054 y(algebr)g(as)34 b Fr(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\))28 b Fr(!)f(P)8 b Fw(\()p Fu(A)23 b Fr(\010)g Fu(C)r(;)17 b(S)27 b Fw(+)22 b Fu(T)14 b Fw(\))p Fv(.)94 4236 y(Pr)-5 b(o)g(of.)42 b Fw(Let)36 b Fu(L)g Fw(denote)h(the)f (closure)g(of)g(the)g(t)m(w)m(o-sided)h(ideal)d(in)h Fu(R)q Fr(hh)p Fu(A)24 b Fr(\010)h Fu(C)7 b Fr(ii)36 b Fw(generated)g(b)m(y)94 4352 y Fu(C)7 b Fw(;)40 b(th)m(us,)g Fu(L)d Fw(is)g(the)h(set)g(of)f(all)e(\(p)s(ossibly)i(in\014nite\))f (linear)g(com)m(binations)g(of)g(paths,)k(eac)m(h)e(of)94 4468 y(whic)m(h)c(con)m(tains)e(at)g(least)h(one)f(arro)m(w)h(from)e Fu(C)7 b Fw(.)44 b(The)33 b(de\014nitions)f(readily)g(imply)f(that)1302 4639 y Fu(R)q Fr(hh)p Fu(A)22 b Fr(\010)h Fu(C)7 b Fr(ii)27 b Fw(=)g Fu(R)q Fr(hh)p Fu(A)p Fr(ii)22 b(\010)g Fu(L;)94 4809 y Fw(and)1418 4938 y Fu(J)9 b Fw(\()p Fu(S)28 b Fw(+)22 b Fu(T)14 b Fw(\))27 b(=)h Fu(J)9 b Fw(\()p Fu(S)d Fw(\))22 b Fr(\010)h Fu(L)94 5088 y Fw(\(in)47 b(the)g(last)g(equalit)m (y)-8 b(,)50 b Fu(J)9 b Fw(\()p Fu(S)d Fw(\))47 b(is)f(understo)s(o)s (d)i(as)f(the)h(Jacobian)e(ideal)f(of)i Fu(S)53 b Fw(in)46 b Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fw(\).)94 5204 y(Therefore,)371 5375 y Fr(P)8 b Fw(\()p Fu(A)23 b Fr(\010)f Fu(C)r(;)17 b(S)28 b Fw(+)22 b Fu(T)14 b Fw(\))27 b(=)g Fu(R)q Fr(hh)p Fu(A)22 b Fr(\010)h Fu(C)7 b Fr(ii)p Fu(=J)i Fw(\()p Fu(S)27 b Fw(+)22 b Fu(T)14 b Fw(\))27 b(=)h(\()p Fu(R)q Fr(hh)p Fu(A)p Fr(ii)21 b(\010)i Fu(L)p Fw(\))p Fu(=)p Fw(\()p Fu(J)9 b Fw(\()p Fu(S)d Fw(\))22 b Fr(\010)g Fu(L)p Fw(\))1119 5498 y Fr(\030)1120 5530 y Fw(=)1224 5526 y Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fu(=J)9 b Fw(\()p Fu(S)d Fw(\))27 b(=)g Fr(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\))p Fu(;)94 5696 y Fw(as)33 b(desired.)3045 b Fh(\003)p eop end end %%Page: 14 14 TeXDict begin HPSdict begin 14 13 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.14) cvn H.B /DEST pdfmark end 94 218 a Fs(14)492 b(HARM)25 b(DERKSEN,)f(JERZY)i(WEYMAN,)f(AND)f(ANDREI)g(ZELEVINSKY)194 420 y Fw(F)-8 b(or)39 b(an)h(arbitrary)e(QP)i(\()p Fu(A;)17 b(S)6 b Fw(\),)41 b(w)m(e)g(denote)g(b)m(y)f Fu(S)2130 384 y Fp(\(2\))2264 420 y Fr(2)h Fu(A)2444 384 y Fp(2)2523 420 y Fw(the)f(degree)h(2)e(homogeneous)94 539 y(comp)s(onen)m(t)j(of)g Fu(S)6 b Fw(.)71 b(W)-8 b(e)43 b(call)d(\()p Fu(A;)17 b(S)6 b Fw(\))41 b Fv(r)-5 b(e)g(duc)g(e)g(d)52 b Fw(if)41 b Fu(S)2078 503 y Fp(\(2\))2216 539 y Fw(=)j(0,)g(i.e.,)g Fu(S)49 b Fr(2)44 b Fl(m)p Fw(\()p Fu(A)p Fw(\))3096 503 y Fp(3)3136 539 y Fw(.)72 b(W)-8 b(e)42 b(de\014ne)94 655 y(the)34 b Fv(trivial)44 b Fw(and)33 b Fv(r)-5 b(e)g(duc)g(e)g(d)43 b Fw(arro)m(w)33 b(spans)i(of)e(\()p Fu(A;)17 b(S)6 b Fw(\))32 b(as)i(the)f(\014nite-dimensional)d Fu(R)q Fw(-bimo)s(dules)94 772 y(giv)m(en)j(b)m(y)94 965 y(\(4.4\))730 965 y SDict begin H.S end 730 965 a 730 965 a SDict begin 14 H.A end 730 965 a 730 965 a SDict begin [ /View [/XYZ H.V] /Dest (equation.4.4) cvn H.B /DEST pdfmark end 730 965 a Fu(A)803 980 y Fp(triv)948 965 y Fw(=)27 b Fu(A)1124 980 y Fp(triv)1241 965 y Fw(\()p Fu(S)6 b Fw(\))28 b(=)f Fu(@)5 b(S)1636 924 y Fp(\(2\))1731 965 y Fu(;)115 b(A)1946 980 y Fp(red)2076 965 y Fw(=)27 b Fu(A)2252 980 y Fp(red)2354 965 y Fw(\()p Fu(S)6 b Fw(\))28 b(=)f Fu(A=@)5 b(S)2871 924 y Fp(\(2\))2999 965 y Fu(:)94 1157 y Fw(\(see)34 b(\()328 1157 y SDict begin H.S end 328 1157 a Fw(4.3)453 1092 y SDict begin H.R end 453 1092 a 453 1157 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.4.3) cvn H.B /ANN pdfmark end 453 1157 a Fw(\)\).)194 1273 y(The)g(follo)m(wing)c(statemen)m(t)j(will)d(pla)m (y)i(a)g(crucial)g(role)f(in)h(later)g(sections.)94 1292 y SDict begin H.S end 94 1292 a 94 1292 a SDict begin 14 H.A end 94 1292 a 94 1292 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.4.6) cvn H.B /DEST pdfmark end 94 1292 a 175 x FA(Theorem)57 b(4.6)f Fw(\(Splitting)47 b(Theorem\))p FA(.)k Fv(F)-7 b(or)49 b(every)h(QP)g Fw(\()p Fu(A;)17 b(S)6 b Fw(\))50 b Fv(with)g(the)g(trivial)g(arr)-5 b(ow)94 1583 y(sp)g(an)48 b Fu(A)405 1598 y Fp(triv)571 1583 y Fv(and)f(the)i(r)-5 b(e)g(duc)g(e)g(d)48 b(arr)-5 b(ow)48 b(sp)-5 b(an)48 b Fu(A)1906 1598 y Fp(red)2008 1583 y Fv(,)k(ther)-5 b(e)48 b(exist)g(a)g(trivial)h(QP)f Fw(\()p Fu(A)3286 1598 y Fp(triv)3403 1583 y Fu(;)17 b(S)3507 1598 y Fp(triv)3623 1583 y Fw(\))94 1699 y Fv(and)39 b(a)h(r)-5 b(e)g(duc)g(e)g(d)39 b(QP)g Fw(\()p Fu(A)1016 1714 y Fp(red)1118 1699 y Fu(;)17 b(S)1222 1714 y Fp(red)1324 1699 y Fw(\))39 b Fv(such)h(that)f Fw(\()p Fu(A;)17 b(S)6 b Fw(\))39 b Fv(is)g(right-e)-5 b(quivalent)39 b(to)h(the)f(dir)-5 b(e)g(ct)40 b(sum)94 1816 y Fw(\()p Fu(A)205 1831 y Fp(triv)322 1816 y Fu(;)17 b(S)426 1831 y Fp(triv)543 1816 y Fw(\))6 b Fr(\010)g Fw(\()p Fu(A)781 1831 y Fp(red)884 1816 y Fu(;)17 b(S)988 1831 y Fp(red)1090 1816 y Fw(\))p Fv(.)42 b(F)-7 b(urthermor)i(e,)28 b(the)g(right-e)-5 b(quivalenc)g(e)26 b(class)h(of)g(e)-5 b(ach)27 b(of)g(the)h(QPs)94 1932 y Fw(\()p Fu(A)205 1947 y Fp(triv)322 1932 y Fu(;)17 b(S)426 1947 y Fp(triv)543 1932 y Fw(\))35 b Fv(and)f Fw(\()p Fu(A)916 1947 y Fp(red)1018 1932 y Fu(;)17 b(S)1122 1947 y Fp(red)1224 1932 y Fw(\))35 b Fv(is)g(determine)-5 b(d)34 b(by)h(the)g(right-e)-5 b(quivalenc)g(e)33 b(class)h(of)h Fw(\()p Fu(A;)17 b(S)6 b Fw(\))p Fv(.)194 2126 y Fw(Let)33 b(us)g(\014rst)g(pro)m(v)m(e)h(the)f(existence)h(of)e(a)g(desired)i (righ)m(t-equiv)-5 b(alence)94 2317 y(\(4.5\))1136 2317 y SDict begin H.S end 1136 2317 a 1136 2317 a SDict begin 14 H.A end 1136 2317 a 1136 2317 a SDict begin [ /View [/XYZ H.V] /Dest (equation.4.5) cvn H.B /DEST pdfmark end 1136 2317 a Fw(\()p Fu(A;)17 b(S)6 b Fw(\))1422 2290 y Fr(\030)1423 2321 y Fw(=)1527 2317 y(\()p Fu(A)1638 2332 y Fp(triv)1755 2317 y Fu(;)17 b(S)1859 2332 y Fp(triv)1976 2317 y Fw(\))22 b Fr(\010)h Fw(\()p Fu(A)2247 2332 y Fp(red)2349 2317 y Fu(;)17 b(S)2453 2332 y Fp(red)2555 2317 y Fw(\))p Fu(:)94 2512 y Fw(There)34 b(is)d(nothing)g(to)h(pro)m(v)m(e)h(if)e(\() p Fu(A;)17 b(S)6 b Fw(\))32 b(is)f(reduced,)j(so)e(let)f(us)i(assume)g (that)e Fu(S)3075 2475 y Fp(\(2\))3197 2512 y Fr(6)p Fw(=)d(0.)43 b(Using)94 2628 y(Prop)s(osition)620 2628 y SDict begin H.S end 620 2628 a Fw(4.4)744 2564 y SDict begin H.R end 744 2564 a 744 2628 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.4) cvn H.B /ANN pdfmark end 744 2628 a 33 w Fw(and)33 b(replacing)f Fu(S)39 b Fw(if)32 b(necessary)j(b)m(y)f(a)f(cyclically)e(equiv)-5 b(alen)m(t)33 b(p)s(oten)m(tial,)e(w)m(e)94 2744 y(can)i(assume)g(that)g Fu(S)38 b Fw(is)32 b(of)g(the)h(form)94 3007 y(\(4.6\))1159 3007 y SDict begin H.S end 1159 3007 a 1159 3007 a SDict begin 14 H.A end 1159 3007 a 1159 3007 a SDict begin [ /View [/XYZ H.V] /Dest (equation.4.6) cvn H.B /DEST pdfmark end 1159 3007 a Fu(S)h Fw(=)1397 2883 y Fq(N)1356 2913 y Fm(X)1364 3125 y Fq(k)r Fp(=1)1500 3007 y Fw(\()p Fu(a)1589 3022 y Fq(k)1632 3007 y Fu(b)1673 3022 y Fq(k)1738 3007 y Fw(+)22 b Fu(a)1887 3022 y Fq(k)1930 3007 y Fu(u)1986 3022 y Fq(k)2050 3007 y Fw(+)g Fu(v)2195 3022 y Fq(k)2238 3007 y Fu(b)2279 3022 y Fq(k)2322 3007 y Fw(\))g(+)g Fu(S)2546 2966 y Fn(0)2570 3007 y Fu(;)94 3287 y Fw(where)32 b(eac)m(h)g Fu(a)643 3302 y Fq(k)686 3287 y Fu(b)727 3302 y Fq(k)801 3287 y Fw(is)e(a)h(cyclic)f(2-path,)h(the)g(arro)m(ws)g Fu(a)2105 3302 y Fp(1)2145 3287 y Fu(;)17 b(b)2230 3302 y Fp(1)2269 3287 y Fu(;)g(:)g(:)g(:)f(;)h(a)2539 3302 y Fq(N)2606 3287 y Fu(;)g(b)2691 3302 y Fq(N)2789 3287 y Fw(form)30 b(a)g(basis)h(of)f Fu(A)3517 3302 y Fp(triv)3634 3287 y Fw(,)94 3404 y(the)23 b(elemen)m(ts)g Fu(u)697 3419 y Fq(k)761 3404 y Fw(and)g Fu(v)988 3419 y Fq(k)1053 3404 y Fw(b)s(elong)e(to)h Fl(m)1539 3367 y Fp(2)1579 3404 y Fw(,)j(and)d(the)h(p)s(oten)m(tial)e Fu(S)2436 3367 y Fn(0)2486 3404 y Fr(2)29 b Fl(m)2657 3367 y Fp(3)2719 3404 y Fw(is)22 b(a)g(linear)e(com)m(bination)94 3520 y(of)43 b(cyclic)g(paths)h(con)m(taining)d(none)j(of)e(the)i(arro)m(ws) g Fu(a)2174 3535 y Fq(k)2260 3520 y Fw(or)e Fu(b)2430 3535 y Fq(k)2473 3520 y Fw(.)76 b(The)44 b(existence)g(of)f(a)g(righ)m (t-)94 3636 y(equiv)-5 b(alence)33 b(\()647 3636 y SDict begin H.S end 647 3636 a Fw(4.5)772 3572 y SDict begin H.R end 772 3572 a 772 3636 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.4.5) cvn H.B /ANN pdfmark end 772 3636 a Fw(\))f(b)s(ecomes)h(a)g(consequence)i(of)d(the)h(follo)m(wing)d (lemma.)94 3661 y SDict begin H.S end 94 3661 a 94 3661 a SDict begin 14 H.A end 94 3661 a 94 3661 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.4.7) cvn H.B /DEST pdfmark end 94 3661 a 169 x FA(Lemma)46 b(4.7.)f Fv(F)-7 b(or)41 b(every)g(p)-5 b(otential)41 b Fu(S)48 b Fv(of)41 b(the)g(form)g Fw(\()2254 3830 y SDict begin H.S end 2254 3830 a Fw(4.6)2379 3766 y SDict begin H.R end 2379 3766 a 2379 3830 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.4.6) cvn H.B /ANN pdfmark end 2379 3830 a Fw(\))p Fv(,)i(ther)-5 b(e)41 b(exists)g(a)h(unitriangular)94 3946 y(automorphism)d Fu(')g Fv(of)g Fu(R)q Fr(hh)p Fu(A)p Fr(ii)g Fv(such)g(that)h Fu(')p Fw(\()p Fu(S)6 b Fw(\))39 b Fv(is)g(cyclic)-5 b(al)5 b(ly)39 b(e)-5 b(quivalent)39 b(to)g(a)h(p)-5 b(otential)39 b(of)94 4063 y(the)c(form)g Fw(\()530 4063 y SDict begin H.S end 530 4063 a Fw(4.6)654 3998 y SDict begin H.R end 654 3998 a 654 4063 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.4.6) cvn H.B /ANN pdfmark end 654 4063 a Fw(\))g Fv(with)g Fu(u)995 4078 y Fq(k)1065 4063 y Fw(=)27 b Fu(v)1215 4078 y Fq(k)1286 4063 y Fw(=)g(0)35 b Fv(for)g(al)5 b(l)34 b Fu(k)s Fv(.)194 4257 y Fw(W)-8 b(e)31 b(sa)m(y)h(that)e(a)g(p)s(oten)m(tial)f Fu(S)37 b Fw(is)30 b Fu(d)p Fv(-split)39 b Fw(if)29 b(it)h(is)g(of)g(the)h (form)f(\()2541 4257 y SDict begin H.S end 2541 4257 a Fw(4.6)2665 4192 y SDict begin H.R end 2665 4192 a 2665 4257 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.4.6) cvn H.B /ANN pdfmark end 2665 4257 a Fw(\))h(with)f Fu(u)3010 4272 y Fq(k)3052 4257 y Fu(;)17 b(v)3143 4272 y Fq(k)3213 4257 y Fr(2)28 b Fl(m)3383 4220 y Fq(d)p Fp(+1)3545 4257 y Fw(for)94 4373 y(all)j Fu(k)s Fw(.)44 b(T)-8 b(o)32 b(pro)m(v)m(e)i(Lemma)1109 4373 y SDict begin H.S end 1109 4373 a Fw(4.7)1234 4309 y SDict begin H.R end 1234 4309 a 1234 4373 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.7) cvn H.B /ANN pdfmark end 1234 4373 a Fw(,)e(w)m(e)i(\014rst)f (sho)m(w)h(the)f(follo)m(wing.)94 4392 y SDict begin H.S end 94 4392 a 94 4392 a SDict begin 14 H.A end 94 4392 a 94 4392 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.4.8) cvn H.B /DEST pdfmark end 94 4392 a 175 x FA(Lemma)j(4.8.)k Fv(Supp)-5 b(ose)33 b(a)g(p)-5 b(otential)33 b Fu(S)39 b Fv(is)33 b Fu(d)p Fv(-split)g(for)g(some)f Fu(d)27 b Fr(\025)i Fw(1)p Fv(.)44 b(Ther)-5 b(e)32 b(exists)h(a)g(unitri-)94 4683 y(angular)k(automorphism)f Fu(')h Fv(of)g Fu(R)q Fr(hh)p Fu(A)p Fr(ii)g Fv(having)f(depth)g Fu(d)h Fv(and)g(such)g(that) g Fu(')p Fw(\()p Fu(S)6 b Fw(\))37 b Fv(is)g(cyclic)-5 b(al)5 b(ly)94 4808 y(e)-5 b(quivalent)35 b(to)g(a)g Fw(2)p Fu(d)p Fv(-split)f(p)-5 b(otential)1511 4782 y Fw(~)1494 4808 y Fu(S)41 b Fv(with)34 b Fu(')p Fw(\()p Fu(S)6 b Fw(\))22 b Fr(\000)2150 4782 y Fw(~)2134 4808 y Fu(S)68 b Fr(2)28 b Fl(m)2432 4771 y Fp(2)p Fq(d)p Fp(+2)2599 4808 y Fv(.)94 5002 y(Pr)-5 b(o)g(of.)42 b Fw(Let)24 b(us)h(write)g Fu(S)30 b Fw(in)24 b(the)h(form)e(\()1534 5002 y SDict begin H.S end 1534 5002 a Fw(4.6)1659 4937 y SDict begin H.R end 1659 4937 a 1659 5002 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.4.6) cvn H.B /ANN pdfmark end 1659 5002 a Fw(\))h(with)g Fu(u)1991 5017 y Fq(k)2033 5002 y Fu(;)17 b(v)2124 5017 y Fq(k)2195 5002 y Fr(2)28 b Fl(m)2365 4965 y Fq(d)p Fp(+1)2496 5002 y Fw(.)40 b(Let)25 b Fu(')g Fw(b)s(e)f(the)h(unitriangular)94 5118 y(automorphism)31 b(of)h Fu(R)q Fr(hh)p Fu(A)p Fr(ii)g Fw(acting)f(on)i(arro)m(ws)g(as)g (follo)m(ws:)255 5309 y Fu(')p Fw(\()p Fu(a)408 5324 y Fq(k)451 5309 y Fw(\))28 b(=)f Fu(a)671 5324 y Fq(k)736 5309 y Fr(\000)c Fu(v)883 5324 y Fq(k)926 5309 y Fu(;)50 b(')p Fw(\()p Fu(b)1146 5324 y Fq(k)1189 5309 y Fw(\))27 b(=)h Fu(b)1399 5324 y Fq(k)1464 5309 y Fr(\000)23 b Fu(u)1620 5324 y Fq(k)1662 5309 y Fu(;)50 b(')p Fw(\()p Fu(c)p Fw(\))27 b(=)h Fu(c)97 b Fw(\()p Fu(c)28 b Fr(2)g Fu(Q)2470 5324 y Fp(1)2532 5309 y Fr(\000)22 b(f)p Fu(a)2732 5324 y Fp(1)2772 5309 y Fu(;)17 b(b)2857 5324 y Fp(1)2896 5309 y Fu(;)g(:)g(:)g(:)f(;)h(a)3166 5324 y Fq(N)3233 5309 y Fu(;)g(b)3318 5324 y Fq(N)3386 5309 y Fr(g)p Fw(\))p Fu(:)94 5501 y Fw(Then)34 b Fu(')f Fw(is)f(of)g(depth)h Fu(d)p Fw(,)g(so)f(b)m(y)i(\()1335 5501 y SDict begin H.S end 1335 5501 a Fw(2.4)1459 5436 y SDict begin H.R end 1459 5436 a 1459 5501 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.2.4) cvn H.B /ANN pdfmark end 1459 5501 a Fw(\),)f(for)f(eac)m(h)h Fu(k)s Fw(,)g(w)m(e)h(ha)m(v)m(e)768 5696 y Fu(')p Fw(\()p Fu(u)926 5711 y Fq(k)968 5696 y Fw(\))28 b(=)f Fu(u)1193 5711 y Fq(k)1258 5696 y Fw(+)22 b Fu(u)1412 5655 y Fn(0)1412 5721 y Fq(k)1454 5696 y Fu(;)50 b(')p Fw(\()p Fu(v)1680 5711 y Fq(k)1722 5696 y Fw(\))28 b(=)g Fu(v)1939 5711 y Fq(k)2004 5696 y Fw(+)22 b Fu(v)2153 5655 y Fn(0)2149 5721 y Fq(k)2289 5696 y Fw(\()p Fu(u)2383 5655 y Fn(0)2383 5721 y Fq(k)2425 5696 y Fu(;)17 b(v)2520 5655 y Fn(0)2516 5721 y Fq(k)2586 5696 y Fr(2)28 b Fl(m)2756 5655 y Fp(2)p Fq(d)p Fp(+1)2923 5696 y Fw(\))p Fu(:)p eop end end %%Page: 15 15 TeXDict begin HPSdict begin 15 14 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.15) cvn H.B /DEST pdfmark end 94 218 a 1179 w Fs(QUIVERS)31 b(WITH)i(POTENTIALS)g(I)1102 b(15)94 417 y Fw(Therefore,)34 b(w)m(e)g(obtain)230 602 y Fu(')p Fw(\()p Fu(S)6 b Fw(\))27 b(=)566 508 y Fm(X)619 720 y Fq(k)710 602 y Fw(\(\()p Fu(a)837 617 y Fq(k)902 602 y Fr(\000)c Fu(v)1049 617 y Fq(k)1092 602 y Fw(\)\()p Fu(b)1209 617 y Fq(k)1274 602 y Fr(\000)f Fu(u)1429 617 y Fq(k)1472 602 y Fw(\))g(+)g(\()p Fu(a)1719 617 y Fq(k)1784 602 y Fr(\000)g Fu(v)1930 617 y Fq(k)1973 602 y Fw(\)\()p Fu(u)2105 617 y Fq(k)2169 602 y Fw(+)g Fu(u)2323 561 y Fn(0)2323 627 y Fq(k)2365 602 y Fw(\))h(+)f(\()p Fu(v)2609 617 y Fq(k)2673 602 y Fw(+)g Fu(v)2822 561 y Fn(0)2818 627 y Fq(k)2861 602 y Fw(\)\()p Fu(b)2978 617 y Fq(k)3043 602 y Fr(\000)h Fu(u)3199 617 y Fq(k)3241 602 y Fw(\)\))f(+)g Fu(S)3503 561 y Fn(0)463 872 y Fw(=)566 777 y Fm(X)619 989 y Fq(k)710 872 y Fw(\()p Fu(a)799 887 y Fq(k)842 872 y Fu(b)883 887 y Fq(k)948 872 y Fw(+)g Fu(a)1097 887 y Fq(k)1140 872 y Fu(u)1196 831 y Fn(0)1196 896 y Fq(k)1261 872 y Fw(+)g Fu(v)1410 831 y Fn(0)1406 896 y Fq(k)1448 872 y Fu(b)1489 887 y Fq(k)1533 872 y Fw(\))g(+)g Fu(S)1751 887 y Fp(1)1812 872 y Fw(+)g Fu(S)1976 831 y Fn(0)1999 872 y Fu(;)94 1130 y Fw(where)1008 1259 y Fu(S)1068 1274 y Fp(1)1135 1259 y Fw(=)28 b Fr(\000)1333 1164 y Fm(X)1385 1376 y Fq(k)1477 1259 y Fw(\()p Fu(v)1562 1274 y Fq(k)1604 1259 y Fu(u)1660 1274 y Fq(k)1725 1259 y Fw(+)22 b Fu(v)1870 1274 y Fq(k)1913 1259 y Fu(u)1969 1217 y Fn(0)1969 1283 y Fq(k)2033 1259 y Fw(+)g Fu(v)2182 1217 y Fn(0)2178 1283 y Fq(k)2221 1259 y Fu(u)2277 1274 y Fq(k)2319 1259 y Fw(\))27 b Fr(2)i Fl(m)2555 1217 y Fp(2)p Fq(d)p Fp(+2)2721 1259 y Fu(:)94 1496 y Fw(In)52 b(view)g(of)e(De\014nition)1081 1496 y SDict begin H.S end 1081 1496 a Fw(3.2)1205 1432 y SDict begin H.R end 1205 1432 a 1205 1496 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.3.2) cvn H.B /ANN pdfmark end 1205 1496 a Fw(,)56 b Fu(S)1348 1511 y Fp(1)1439 1496 y Fw(is)51 b(cyclically)e(equiv)-5 b(alen)m(t)51 b(to)g(a)g(p)s(oten)m(tial)e(of)i(the)h(form)94 1538 y Fm(P)200 1642 y Fq(k)242 1613 y Fw(\()p Fu(a)331 1628 y Fq(k)374 1613 y Fu(u)430 1576 y Fn(00)430 1638 y Fq(k)504 1613 y Fw(+)31 b Fu(v)662 1576 y Fn(00)658 1638 y Fq(k)704 1613 y Fu(b)745 1628 y Fq(k)788 1613 y Fw(\))g(+)h Fu(S)1031 1576 y Fn(00)1073 1613 y Fw(,)49 b(where)e Fu(u)1500 1576 y Fn(00)1500 1638 y Fq(k)1543 1613 y Fu(;)17 b(v)1638 1576 y Fn(00)1634 1638 y Fq(k)1730 1613 y Fr(2)51 b Fl(m)1923 1576 y Fp(2)p Fq(d)p Fp(+1)2089 1613 y Fw(,)f(and)c Fu(S)2435 1576 y Fn(00)2523 1613 y Fw(is)g(a)f(linear)g(com)m(bination)f(of)94 1729 y(cyclic)33 b(paths)g(con)m(taining)e(none)i(of)f(the)h Fu(a)1666 1744 y Fq(k)1742 1729 y Fw(or)f Fu(b)1902 1744 y Fq(k)1945 1729 y Fw(.)43 b(F)-8 b(urthermore,)32 b(w)m(e)i(ha)m(v)m(e)1102 1914 y Fu(S)1162 1929 y Fp(1)1224 1914 y Fr(\000)22 b Fu(S)1389 1873 y Fn(00)1454 1914 y Fr(\000)1553 1819 y Fm(X)1606 2032 y Fq(k)1697 1914 y Fw(\()p Fu(a)1786 1929 y Fq(k)1829 1914 y Fu(u)1885 1873 y Fn(00)1885 1939 y Fq(k)1949 1914 y Fw(+)g Fu(v)2098 1873 y Fn(00)2094 1939 y Fq(k)2141 1914 y Fu(b)2182 1929 y Fq(k)2225 1914 y Fw(\))27 b Fr(2)h Fl(m)2460 1873 y Fp(2)p Fq(d)p Fp(+2)2627 1914 y Fu(:)94 2194 y Fw(W)-8 b(e)33 b(see)h(that)f(the)g(desired)g(p)s (oten)m(tial)1559 2169 y(~)1543 2194 y Fu(S)38 b Fw(can)33 b(b)s(e)f(c)m(hosen)j(as)770 2355 y(~)753 2380 y Fu(S)f Fw(=)950 2285 y Fm(X)1003 2497 y Fq(k)1094 2380 y Fw(\()p Fu(a)1183 2395 y Fq(k)1226 2380 y Fu(b)1267 2395 y Fq(k)1332 2380 y Fw(+)22 b Fu(a)1481 2395 y Fq(k)1524 2380 y Fw(\()p Fu(u)1618 2339 y Fn(0)1618 2404 y Fq(k)1682 2380 y Fw(+)g Fu(u)1836 2339 y Fn(00)1836 2404 y Fq(k)1879 2380 y Fw(\))g(+)g(\()p Fu(v)2126 2339 y Fn(0)2122 2404 y Fq(k)2187 2380 y Fw(+)g Fu(v)2336 2339 y Fn(00)2332 2404 y Fq(k)2378 2380 y Fw(\))p Fu(b)2457 2395 y Fq(k)2500 2380 y Fw(\))g(+)g Fu(S)2724 2339 y Fn(0)2769 2380 y Fw(+)g Fu(S)2933 2339 y Fn(00)2975 2380 y Fu(;)94 2638 y Fw(completing)31 b(the)i(pro)s(of)f(of)g(Lemma) 1475 2638 y SDict begin H.S end 1475 2638 a Fw(4.8)1600 2574 y SDict begin H.R end 1600 2574 a 1600 2638 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.8) cvn H.B /ANN pdfmark end 1600 2638 a Fw(.)1957 b Fh(\003)94 2819 y Fv(Pr)-5 b(o)g(of)35 b(of)g(L)-5 b(emma)816 2838 y SDict begin H.S end 816 2838 a -19 x Fv(4.7)945 2754 y SDict begin H.R end 945 2754 a 945 2819 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.7) cvn H.B /ANN pdfmark end 945 2819 a Fv(.)42 b Fw(Starting)28 b(with)h(a)g(p)s(oten)m(tial)e Fu(S)35 b Fw(of)29 b(the)h(form)e(\()2731 2819 y SDict begin H.S end 2731 2819 a Fw(4.6)2855 2755 y SDict begin H.R end 2855 2755 a 2855 2819 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.4.6) cvn H.B /ANN pdfmark end 2855 2819 a Fw(\))h(and)h(using)f(rep)s(eat-)94 2935 y(edly)i(Lemma)647 2935 y SDict begin H.S end 647 2935 a Fw(4.8)772 2871 y SDict begin H.R end 772 2871 a 772 2935 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.8) cvn H.B /ANN pdfmark end 772 2935 a Fw(,)g(w)m(e)h(construct)g(a)e(sequence)j(of)d(p)s(oten)m (tials)g Fu(S)2498 2950 y Fp(1)2537 2935 y Fu(;)17 b(S)2641 2950 y Fp(2)2680 2935 y Fu(;)g(:)g(:)g(:)f Fw(,)31 b(and)g(a)f (sequence)k(of)94 3052 y(unitriangular)c(automorphisms)h Fu(')1433 3067 y Fp(1)1472 3052 y Fu(;)17 b(')1580 3067 y Fp(2)1620 3052 y Fu(;)g(:)g(:)g(:)e Fw(,)33 b(with)f(the)h(follo)m (wing)d(prop)s(erties:)94 3097 y SDict begin H.S end 94 3097 a 94 3097 a SDict begin 14 H.A end 94 3097 a 94 3097 a SDict begin [ /View [/XYZ H.V] /Dest (Item.1) cvn H.B /DEST pdfmark end 94 3097 a 243 3194 a Fw(\(1\))41 b Fu(S)469 3209 y Fp(1)536 3194 y Fw(=)28 b Fu(S)6 b Fw(.)94 3219 y SDict begin H.S end 94 3219 a 94 3219 a SDict begin 14 H.A end 94 3219 a 94 3219 a SDict begin [ /View [/XYZ H.V] /Dest (Item.2) cvn H.B /DEST pdfmark end 94 3219 a 243 3311 a Fw(\(2\))41 b Fu(S)469 3326 y Fq(d)542 3311 y Fw(is)32 b(2)689 3274 y Fq(d)p Fn(\000)p Fp(1)820 3311 y Fw(-split.)94 3335 y SDict begin H.S end 94 3335 a 94 3335 a SDict begin 14 H.A end 94 3335 a 94 3335 a SDict begin [ /View [/XYZ H.V] /Dest (Item.3) cvn H.B /DEST pdfmark end 94 3335 a 243 3427 a Fw(\(3\))41 b Fu(')473 3442 y Fq(d)546 3427 y Fw(is)32 b(of)g(depth)i(2)1081 3391 y Fq(d)p Fn(\000)p Fp(1)1211 3427 y Fw(.)94 3452 y SDict begin H.S end 94 3452 a 94 3452 a SDict begin 14 H.A end 94 3452 a 94 3452 a SDict begin [ /View [/XYZ H.V] /Dest (Item.4) cvn H.B /DEST pdfmark end 94 3452 a 243 3554 a Fw(\(4\))41 b Fu(')473 3569 y Fq(d)513 3554 y Fw(\()p Fu(S)611 3569 y Fq(d)652 3554 y Fw(\))32 b(is)g(cyclically)f (equiv)-5 b(alen)m(t)32 b(to)h Fu(S)1886 3569 y Fq(d)p Fp(+1)2016 3554 y Fw(,)g(and)g Fu(')2330 3569 y Fq(d)2370 3554 y Fw(\()p Fu(S)2468 3569 y Fq(d)2508 3554 y Fw(\))22 b Fr(\000)h Fu(S)2728 3569 y Fq(d)p Fp(+1)2919 3554 y Fr(2)28 b Fl(m)3089 3518 y Fp(2)3124 3495 y Fg(d)3161 3518 y Fp(+2)3255 3554 y Fw(.)194 3697 y(By)33 b(prop)s(ert)m(y)h (\(3\),)e(setting)94 3866 y(\(4.7\))1384 3866 y SDict begin H.S end 1384 3866 a 1384 3866 a SDict begin 14 H.A end 1384 3866 a 1384 3866 a SDict begin [ /View [/XYZ H.V] /Dest (equation.4.7) cvn H.B /DEST pdfmark end 1384 3866 a Fu(')c Fw(=)52 b(lim)1579 3926 y Fq(n)p Fn(!1)1780 3866 y Fu(')1844 3881 y Fq(n)1891 3866 y Fu(')1955 3881 y Fq(n)p Fn(\000)p Fp(1)2108 3866 y Fr(\001)17 b(\001)g(\001)e Fu(')2305 3881 y Fp(1)2345 3866 y Fu(;)94 4072 y Fw(w)m(e)30 b(obtain)d(a)h(w)m(ell)f(de\014ned)j(unitriangular)25 b(automorphism)h Fu(')i Fw(of)g Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fw(;)g(indeed,)i(in)d(view)i(of)94 4206 y(\()132 4206 y SDict begin H.S end 132 4206 a Fw(2.4)257 4142 y SDict begin H.R end 257 4142 a 257 4206 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.2.4) cvn H.B /ANN pdfmark end 257 4206 a Fw(\),)35 b(for)f(an)m(y)g Fu(u)d Fr(2)g Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fw(,)j(if)f(w)m(e)i(write)f Fu(')1793 4221 y Fq(n)1840 4206 y Fu(')1904 4221 y Fq(n)p Fn(\000)p Fp(1)2058 4206 y Fr(\001)17 b(\001)g(\001)d Fu(')2254 4221 y Fp(1)2294 4206 y Fw(\()p Fu(u)p Fw(\))33 b(as)2581 4131 y Fm(P)2686 4158 y Fn(1)2686 4235 y Fq(d)p Fp(=0)2833 4206 y Fu(u)2889 4155 y Fp(\()p Fq(d)p Fp(\))2889 4216 y Fq(n)3018 4206 y Fw(with)h Fu(u)3298 4155 y Fp(\()p Fq(d)p Fp(\))3298 4216 y Fq(n)3424 4206 y Fr(2)d Fu(A)3594 4170 y Fq(d)3634 4206 y Fw(,)94 4344 y(then)j(eac)m(h)f(homogeneous)g (comp)s(onen)m(t)f Fu(u)1687 4293 y Fp(\()p Fq(d)p Fp(\))1687 4354 y Fq(n)1815 4344 y Fw(stabilizes)f(as)i Fu(n)28 b Fr(!)f(1)p Fw(.)194 4460 y(W)-8 b(e)26 b(claim)d(that)i(this)g (automorphism)e Fu(')j Fw(satis\014es)g(the)g(required)g(prop)s(erties) f(in)g(Lemma)3510 4460 y SDict begin H.S end 3510 4460 a Fw(4.7)3634 4396 y SDict begin H.R end 3634 4396 a 3634 4460 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.7) cvn H.B /ANN pdfmark end 3634 4460 a Fw(.)94 4577 y(T)-8 b(o)32 b(see)g(this,)f(for)g Fu(d)c Fr(\025)h Fw(1,)k(denote)f Fu(C)1430 4592 y Fq(d)1498 4577 y Fw(=)d Fu(')1666 4592 y Fq(d)1706 4577 y Fw(\()p Fu(S)1804 4592 y Fq(d)1845 4577 y Fw(\))19 b Fr(\000)h Fu(S)2059 4592 y Fq(d)p Fp(+1)2189 4577 y Fw(.)43 b(By)32 b(\(4\),)f Fu(C)2664 4592 y Fq(d)2732 4577 y Fr(2)d(f)p Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fu(;)17 b(R)q Fr(hh)p Fu(A)p Fr(iig)g(\\)94 4704 y Fl(m)170 4668 y Fp(2)205 4644 y Fg(d)242 4668 y Fp(+2)375 4704 y Fw(\(recall)37 b(from)h(De\014nition)1375 4704 y SDict begin H.S end 1375 4704 a Fw(3.4)1500 4640 y SDict begin H.R end 1500 4640 a 1500 4704 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.3.4) cvn H.B /ANN pdfmark end 1500 4704 a 39 w Fw(that)g Fr(f)p Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fu(;)17 b(R)q Fr(hh)p Fu(A)p Fr(iig)36 b Fw(denotes)k(the)f(closure)g(of)f(the)94 4820 y(v)m(ector)33 b(subspace)h(in)d Fu(R)q Fr(hh)p Fu(A)p Fr(ii)g Fw(spanned)i(b)m(y)f(all)e(comm)m(utators\).)42 b(Using)32 b(\(1\),)f(it)g(is)g(easy)i(to)e(see)94 4936 y(that)799 5120 y Fu(')863 5135 y Fq(n)910 5120 y Fu(')974 5135 y Fq(n)p Fn(\000)p Fp(1)1128 5120 y Fr(\001)17 b(\001)g(\001)e Fu(')1325 5135 y Fp(1)1364 5120 y Fw(\()p Fu(S)6 b Fw(\))27 b(=)h Fu(S)1697 5135 y Fq(n)p Fp(+1)1856 5120 y Fw(+)2005 4996 y Fq(n)1954 5026 y Fm(X)1963 5238 y Fq(d)p Fp(=1)2115 5120 y Fu(')2179 5135 y Fq(n)2226 5120 y Fu(')2290 5135 y Fq(n)p Fn(\000)p Fp(1)2443 5120 y Fr(\001)17 b(\001)g(\001)e Fu(')2640 5135 y Fq(d)p Fp(+1)2771 5120 y Fw(\()p Fu(C)2879 5135 y Fq(d)2919 5120 y Fw(\))94 5358 y(for)33 b(ev)m(ery)h Fu(n)28 b Fr(\025)g Fw(1;)k(passing)h(to)f(the)h(limit)c(as)k Fu(n)28 b Fr(!)f(1)32 b Fw(yields)997 5602 y Fu(')p Fw(\()p Fu(S)6 b Fw(\))27 b(=)52 b(lim)1334 5662 y Fq(n)p Fn(!1)1534 5602 y Fu(S)1594 5617 y Fq(n)1664 5602 y Fw(+)22 b Fu(')p Fw(\()1900 5477 y Fn(1)1864 5507 y Fm(X)1873 5719 y Fq(d)p Fp(=1)2007 5602 y Fw(\()p Fu(')2109 5617 y Fq(d)2166 5602 y Fr(\001)17 b(\001)g(\001)e Fu(')2363 5617 y Fp(1)2402 5602 y Fw(\))2440 5561 y Fn(\000)p Fp(1)2535 5602 y Fw(\()p Fu(C)2643 5617 y Fq(d)2683 5602 y Fw(\)\))p eop end end %%Page: 16 16 TeXDict begin HPSdict begin 16 15 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.16) cvn H.B /DEST pdfmark end 94 218 a Fs(16)492 b(HARM)25 b(DERKSEN,)f(JERZY)i(WEYMAN,)f(AND)f(ANDREI)g(ZELEVINSKY)94 417 y Fw(\(the)k(con)m(v)m(ergence)i(of)d(the)h(series)g(on)g(the)g (righ)m(t)e(is)h(clear)g(since)h(an)m(y)g(automorphism)d(of)i Fu(R)q Fr(hh)p Fu(A)p Fr(ii)94 533 y Fw(preserv)m(es)55 b(the)e(p)s(o)m(w)m(ers)g(of)f Fl(m)p Fw(\).)102 b(W)-8 b(e)52 b(conclude)h(that)f Fu(')p Fw(\()p Fu(S)6 b Fw(\))51 b(is)h(cyclically)e(equiv)-5 b(alen)m(t)52 b(to)94 649 y(lim)230 664 y Fq(n)p Fn(!1)435 649 y Fu(S)495 664 y Fq(n)542 649 y Fw(.)e(In)35 b(view)g(of)f(\(2\),)h(the)h(latter)d (elemen)m(t)i(is)f(of)h(the)g(form)e(\()2725 649 y SDict begin H.S end 2725 649 a Fw(4.6)2850 585 y SDict begin H.R end 2850 585 a 2850 649 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.4.6) cvn H.B /ANN pdfmark end 2850 649 a Fw(\))i(with)f Fu(u)3203 664 y Fq(k)3277 649 y Fw(=)d Fu(v)3431 664 y Fq(k)3505 649 y Fw(=)h(0)94 766 y(for)38 b(all)e Fu(k)s Fw(.)59 b(This)38 b(completes)g(the)g(pro)s(ofs)f(of)h (Lemma)2155 766 y SDict begin H.S end 2155 766 a Fw(4.7)2280 701 y SDict begin H.R end 2280 701 a 2280 766 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.7) cvn H.B /ANN pdfmark end 2280 766 a 38 w Fw(and)g(of)f(the)h(existence)i(of)d(a)h (righ)m(t-)94 882 y(equiv)-5 b(alence)33 b(\()647 882 y SDict begin H.S end 647 882 a Fw(4.5)772 818 y SDict begin H.R end 772 818 a 772 882 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.4.5) cvn H.B /ANN pdfmark end 772 882 a Fw(\).)2747 b Fh(\003)194 1051 y Fw(The)31 b(ab)s(o)m(v)m(e)g (argumen)m(t)f(mak)m(es)h(it)e(clear)h(that)g(the)h(righ)m(t-equiv)-5 b(alence)29 b(class)h(of)g(\()p Fu(A)3286 1066 y Fp(triv)3403 1051 y Fu(;)17 b(S)3507 1066 y Fp(triv)3623 1051 y Fw(\))94 1167 y(is)42 b(determined)f(b)m(y)h(the)g(righ)m(t-equiv)-5 b(alence)41 b(class)g(of)h(\()p Fu(A;)17 b(S)6 b Fw(\))40 b(.)71 b(T)-8 b(o)41 b(pro)m(v)m(e)i(Theorem)3401 1167 y SDict begin H.S end 3401 1167 a Fw(4.6)3526 1103 y SDict begin H.R end 3526 1103 a 3526 1167 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.6) cvn H.B /ANN pdfmark end 3526 1167 a Fw(,)g(it)94 1283 y(remains)28 b(to)h(sho)m(w)g(that)g (the)g(same)f(is)h(true)g(for)f(\()p Fu(A)1972 1298 y Fp(red)2074 1283 y Fu(;)17 b(S)2178 1298 y Fp(red)2280 1283 y Fw(\).)42 b(Changing)28 b(notation)g(a)g(little)e(bit,)94 1399 y(w)m(e)34 b(need)g(to)e(pro)m(v)m(e)i(the)f(follo)m(wing.)94 1419 y SDict begin H.S end 94 1419 a 94 1419 a SDict begin 14 H.A end 94 1419 a 94 1419 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.4.9) cvn H.B /DEST pdfmark end 94 1419 a 156 x FA(Prop)s(osition)40 b(4.9.)k Fv(L)-5 b(et)38 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\))38 b Fv(and)f Fw(\()p Fu(A;)17 b(S)1803 1539 y Fn(0)1826 1575 y Fw(\))38 b Fv(b)-5 b(e)38 b(r)-5 b(e)g(duc)g(e)g(d)37 b(QPs,)i(and)e Fw(\()p Fu(C)r(;)17 b(T)d Fw(\))37 b Fv(a)h(trivial)g(QP.)94 1691 y(If)32 b Fw(\()p Fu(A)15 b Fr(\010)g Fu(C)r(;)i(S)j Fw(+)15 b Fu(T)f Fw(\))31 b Fv(is)g(right-e)-5 b(quivalent)31 b(to)h Fw(\()p Fu(A)15 b Fr(\010)g Fu(C)r(;)i(S)2137 1655 y Fn(0)2174 1691 y Fw(+)e Fu(T)f Fw(\))31 b Fv(then)h Fw(\()p Fu(A;)17 b(S)6 b Fw(\))31 b Fv(is)g(right-e)-5 b(quivalent)94 1807 y(to)36 b Fw(\()p Fu(A;)17 b(S)433 1771 y Fn(0)455 1807 y Fw(\))p Fv(.)194 1983 y Fw(W)-8 b(e)33 b(deduce)h(Prop)s(osition)1212 1983 y SDict begin H.S end 1212 1983 a Fw(4.9)1337 1919 y SDict begin H.R end 1337 1919 a 1337 1983 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.9) cvn H.B /ANN pdfmark end 1337 1983 a 32 w Fw(from)e(the)h(follo)m(wing)c(result)k(of)f(indep)s(enden)m(t)i (in)m(terest.)94 2002 y SDict begin H.S end 94 2002 a 94 2002 a SDict begin 14 H.A end 94 2002 a 94 2002 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.4.10) cvn H.B /DEST pdfmark end 94 2002 a 157 x FA(Prop)s(osition)e(4.10.)39 b Fv(L)-5 b(et)32 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\))32 b Fv(and)f Fw(\()p Fu(A;)17 b(S)1828 2122 y Fn(0)1851 2159 y Fw(\))31 b Fv(b)-5 b(e)32 b(r)-5 b(e)g(duc)g(e)g(d)31 b(QPs)g(such)h(that)g Fu(S)3069 2122 y Fn(0)3107 2159 y Fr(\000)15 b Fu(S)34 b Fr(2)28 b Fu(J)9 b Fw(\()p Fu(S)d Fw(\))3592 2122 y Fp(2)3632 2159 y Fv(.)94 2275 y(Then)35 b(we)f(have:)94 2296 y SDict begin H.S end 94 2296 a 94 2296 a SDict begin 14 H.A end 94 2296 a 94 2296 a SDict begin [ /View [/XYZ H.V] /Dest (Item.5) cvn H.B /DEST pdfmark end 94 2296 a 243 2413 a Fw(\(1\))41 b Fu(J)9 b Fw(\()p Fu(S)576 2376 y Fn(0)599 2413 y Fw(\))28 b(=)f Fu(J)9 b Fw(\()p Fu(S)d Fw(\))p Fv(.)94 2437 y SDict begin H.S end 94 2437 a 94 2437 a SDict begin 14 H.A end 94 2437 a 94 2437 a SDict begin [ /View [/XYZ H.V] /Dest (Item.6) cvn H.B /DEST pdfmark end 94 2437 a 243 2529 a Fw(\(2\))41 b(\()p Fu(A;)17 b(S)6 b Fw(\))41 b Fv(is)f(right-e)-5 b(quivalent)41 b(to)g Fw(\()p Fu(A;)17 b(S)1856 2493 y Fn(0)1879 2529 y Fw(\))p Fv(.)63 b(Mor)-5 b(e)41 b(pr)-5 b(e)g(cisely,)42 b(ther)-5 b(e)41 b(exists)g(an)f(algebr)-5 b(a)409 2645 y(automorphism)34 b Fu(')h Fv(of)f Fu(R)q Fr(hh)p Fu(A)p Fr(ii)g Fv(such)h(that)g Fu(')p Fw(\()p Fu(S)6 b Fw(\))34 b Fv(is)h(cyclic)-5 b(al)5 b(ly)34 b(e)-5 b(quivalent)34 b(to)h Fu(S)3419 2609 y Fn(0)3442 2645 y Fv(,)g(and)409 2761 y Fu(')p Fw(\()p Fu(u)p Fw(\))21 b Fr(\000)i Fu(u)k Fr(2)h Fu(J)9 b Fw(\()p Fu(S)d Fw(\))35 b Fv(for)g(al)5 b(l)34 b Fu(u)27 b Fr(2)i Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fv(.)94 2937 y(Pr)-5 b(o)g(of.)42 b Fw(\(1\))35 b(Since)i(\()p Fu(A;)17 b(S)6 b Fw(\))35 b(is)h(reduced,)i(w)m(e)f(ha)m (v)m(e)h Fu(J)9 b Fw(\()p Fu(S)d Fw(\))33 b Fr(\022)h Fl(m)2403 2901 y Fp(2)2443 2937 y Fw(.)54 b(As)37 b(an)f(easy)h (consequence)i(of)94 3053 y(the)33 b(cyclic)g(Leibniz)f(rule)g(\()1107 3053 y SDict begin H.S end 1107 3053 a Fw(3.4)1231 2989 y SDict begin H.R end 1231 2989 a 1231 3053 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.3.4) cvn H.B /ANN pdfmark end 1231 3053 a Fw(\),)h(w)m(e)g(see)h(that)1214 3213 y Fu(@)1265 3228 y Fq(\030)1303 3213 y Fw(\()p Fu(J)9 b Fw(\()p Fu(S)d Fw(\))1546 3172 y Fp(2)1585 3213 y Fw(\))1623 3228 y Fp(cyc)1755 3213 y Fr(\022)28 b Fl(m)p Fu(J)9 b Fw(\()p Fu(S)d Fw(\))22 b(+)g Fu(J)9 b Fw(\()p Fu(S)d Fw(\))p Fl(m)94 3372 y Fw(for)33 b(an)m(y)g Fu(\030)f Fr(2)c Fu(A)670 3336 y Fq(?)710 3372 y Fw(.)43 b(It)33 b(follo)m(ws)e(that)94 3531 y(\(4.8\))1235 3531 y SDict begin H.S end 1235 3531 a 1235 3531 a SDict begin 14 H.A end 1235 3531 a 1235 3531 a SDict begin [ /View [/XYZ H.V] /Dest (equation.4.8) cvn H.B /DEST pdfmark end 1235 3531 a Fu(@)1286 3546 y Fq(\030)1324 3531 y Fu(S)1390 3490 y Fn(0)1435 3531 y Fr(\000)23 b Fu(@)1586 3546 y Fq(\030)1624 3531 y Fu(S)34 b Fr(2)28 b Fl(m)p Fu(J)9 b Fw(\()p Fu(S)d Fw(\))22 b(+)g Fu(J)9 b Fw(\()p Fu(S)d Fw(\))p Fl(m)p Fu(;)94 3690 y Fw(implying)30 b(that)j Fu(J)9 b Fw(\()p Fu(S)877 3654 y Fn(0)900 3690 y Fw(\))27 b Fr(\022)i Fu(J)9 b Fw(\()p Fu(S)d Fw(\).)194 3807 y(T)-8 b(o)33 b(sho)m(w)g(the)g(rev)m(erse)i(inclusion,)c(note)i(that)f(\()1973 3807 y SDict begin H.S end 1973 3807 a Fw(4.8)2098 3742 y SDict begin H.R end 2098 3742 a 2098 3807 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.4.8) cvn H.B /ANN pdfmark end 2098 3807 a Fw(\))g(also)g(implies)e(that)1142 3966 y Fu(J)9 b Fw(\()p Fu(S)d Fw(\))28 b Fr(\022)g Fu(J)9 b Fw(\()p Fu(S)1647 3925 y Fn(0)1670 3966 y Fw(\))22 b(+)g(\()p Fl(m)p Fu(J)9 b Fw(\()p Fu(S)d Fw(\))22 b(+)g Fu(J)9 b Fw(\()p Fu(S)d Fw(\))p Fl(m)p Fw(\))p Fu(:)94 4125 y Fw(Applying)32 b(the)h(same)g(inclusion)e(to)h(eac)m(h)h(of)f (the)h(terms)g Fu(J)9 b Fw(\()p Fu(S)d Fw(\))32 b(on)h(the)g(righ)m(t,) f(w)m(e)h(obtain)864 4285 y Fu(J)9 b Fw(\()p Fu(S)d Fw(\))27 b Fr(\022)i Fu(J)9 b Fw(\()p Fu(S)1369 4244 y Fn(0)1392 4285 y Fw(\))22 b(+)g(\()p Fl(m)1664 4244 y Fp(2)1704 4285 y Fu(J)9 b Fw(\()p Fu(S)d Fw(\))22 b(+)g Fl(m)p Fu(J)9 b Fw(\()p Fu(S)d Fw(\))p Fl(m)22 b Fw(+)g Fu(J)9 b Fw(\()p Fu(S)d Fw(\))p Fl(m)2787 4244 y Fp(2)2827 4285 y Fw(\))p Fu(:)94 4444 y Fw(Con)m(tin)m(uing)32 b(in)g(the)h(same)g(w)m (a)m(y)-8 b(,)33 b(w)m(e)h(get)837 4678 y Fu(J)9 b Fw(\()p Fu(S)d Fw(\))28 b Fr(\022)g Fu(J)9 b Fw(\()p Fu(S)1342 4636 y Fn(0)1365 4678 y Fw(\))22 b(+)1574 4553 y Fq(n)1523 4583 y Fm(X)1531 4795 y Fq(k)r Fp(=0)1684 4678 y Fl(m)1760 4636 y Fq(k)1803 4678 y Fu(J)9 b Fw(\()p Fu(S)d Fw(\))p Fl(m)2084 4636 y Fq(n)p Fn(\000)p Fq(k)2252 4678 y Fr(\022)28 b Fu(J)9 b Fw(\()p Fu(S)2524 4636 y Fn(0)2547 4678 y Fw(\))22 b(+)g Fl(m)2781 4636 y Fq(n)p Fp(+2)94 4931 y Fw(for)28 b(an)m(y)i Fu(n)d Fr(\025)i Fw(1.)41 b(Remem)m(b)s(ering)28 b(the)g(de\014nition)g(of)g(top)s(ology)f(in)g Fu(R)q Fr(hh)p Fu(A)p Fr(ii)h Fw(\(see)i(\()3112 4931 y SDict begin H.S end 3112 4931 a Fw(2.3)3236 4867 y SDict begin H.R end 3236 4867 a 3236 4931 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.2.3) cvn H.B /ANN pdfmark end 3236 4931 a Fw(\)\))e(and)h(the)94 5047 y(fact)k(that)g Fu(J)9 b Fw(\()p Fu(S)666 5011 y Fn(0)689 5047 y Fw(\))33 b(is)f(closed,)h(w)m (e)h(conclude)g(that)e Fu(J)9 b Fw(\()p Fu(S)d Fw(\))28 b Fr(\022)h Fu(J)9 b Fw(\()p Fu(S)2436 5011 y Fn(0)2459 5047 y Fw(\),)33 b(\014nishing)f(the)h(pro)s(of)f(of)h(part)94 5163 y(\(1\))g(of)f(Prop)s(osition)888 5163 y SDict begin H.S end 888 5163 a Fw(4.10)1061 5099 y SDict begin H.R end 1061 5099 a 1061 5163 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.10) cvn H.B /ANN pdfmark end 1061 5163 a Fw(.)194 5280 y(\(2\))45 b(Let)g Fu(Q)628 5295 y Fp(1)716 5280 y Fw(=)j Fr(f)p Fu(a)941 5295 y Fp(1)981 5280 y Fu(;)17 b(:)g(:)g(:)f(;)h(a)1251 5295 y Fq(N)1318 5280 y Fr(g)44 b Fw(b)s(e)h(the)h(set)f(of)g(arro)m(ws)g(\(that)g(is,)i(a)e (basis)g(of)f Fu(A)p Fw(\).)80 b(Then)94 5396 y(a)45 b(unitriangular)e(automorphism)f Fu(')j Fw(of)g Fu(R)q Fr(hh)p Fu(A)p Fr(ii)f Fw(is)h(sp)s(eci\014ed)h(b)m(y)g(a)f Fu(N)10 b Fw(-tuple)44 b(of)h(elemen)m(ts)94 5512 y Fu(b)135 5527 y Fp(1)175 5512 y Fu(;)17 b(:)g(:)g(:)f(;)h(b)435 5527 y Fq(N)530 5512 y Fr(2)28 b Fl(m)700 5476 y Fp(2)773 5512 y Fw(suc)m(h)34 b(that)94 5671 y(\(4.9\))1208 5671 y SDict begin H.S end 1208 5671 a 1208 5671 a SDict begin 14 H.A end 1208 5671 a 1208 5671 a SDict begin [ /View [/XYZ H.V] /Dest (equation.4.9) cvn H.B /DEST pdfmark end 1208 5671 a Fu(')p Fw(\()p Fu(a)1361 5686 y Fq(k)1404 5671 y Fw(\))27 b(=)h Fu(a)1624 5686 y Fq(k)1689 5671 y Fw(+)22 b Fu(b)1828 5686 y Fq(k)1904 5671 y Fw(\()p Fu(k)31 b Fw(=)c(1)p Fu(;)17 b(:)g(:)g(:)f(;)h(N)10 b Fw(\))p Fu(:)94 5739 y SDict begin H.S end 94 5739 a 94 5739 a SDict begin 14 H.A end 94 5739 a 94 5739 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.4.11) cvn H.B /DEST pdfmark end 94 5739 a eop end end %%Page: 17 17 TeXDict begin HPSdict begin 17 16 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.17) cvn H.B /DEST pdfmark end 94 218 a 1179 w Fs(QUIVERS)31 b(WITH)i(POTENTIALS)g(I)1102 b(17)94 417 y FA(Lemma)44 b(4.11.)h Fv(L)-5 b(et)40 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\))40 b Fv(b)-5 b(e)40 b(a)g(r)-5 b(e)g(duc)g(e)g(d)39 b(QP,)h(and)g(let)g Fu(')g Fv(b)-5 b(e)40 b(a)g(unitriangular)g(automor-)94 544 y(phism)48 b(of)g Fu(R)q Fr(hh)p Fu(A)p Fr(ii)g Fv(given)g(by)g Fw(\()1318 544 y SDict begin H.S end 1318 544 a Fw(4.9)1443 480 y SDict begin H.R end 1443 480 a 1443 544 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.4.9) cvn H.B /ANN pdfmark end 1443 544 a Fw(\))p Fv(.)86 b(Then)47 b(the)i(p)-5 b(otential)48 b Fu(')p Fw(\()p Fu(S)6 b Fw(\))32 b Fr(\000)h Fu(S)38 b Fr(\000)3007 469 y Fm(P)3113 496 y Fq(N)3113 573 y(k)r Fp(=1)3262 544 y Fu(b)3303 559 y Fq(k)3346 544 y Fu(@)3397 559 y Fq(a)3434 571 y Fg(k)3477 544 y Fu(S)55 b Fv(is)94 660 y(cyclic)-5 b(al)5 b(ly)47 b(e)-5 b(quivalent)47 b(to)g(an)g(element)f(of)h Fl(m)p Fu(I)1899 624 y Fp(2)1938 660 y Fv(,)j(wher)-5 b(e)47 b Fu(I)55 b Fv(is)47 b(the)g(closur)-5 b(e)46 b(of)h(the)g(ide)-5 b(al)47 b(in)94 777 y Fu(R)q Fr(hh)p Fu(A)p Fr(ii)35 b Fv(gener)-5 b(ate)g(d)34 b(by)h Fu(b)1033 792 y Fp(1)1073 777 y Fu(;)17 b(:)g(:)g(:)e(;)i(b)1332 792 y Fq(N)1400 777 y Fv(.)94 951 y(Pr)-5 b(o)g(of.)42 b Fw(First)32 b(consider)i(the)g(case)g(where)h Fu(S)g Fw(=)29 b Fu(a)1922 966 y Fq(k)1959 975 y Fo(1)2014 951 y Fr(\001)17 b(\001)g(\001)e Fu(a)2198 966 y Fq(k)2235 978 y Fg(d)2309 951 y Fw(is)33 b(a)g(cyclic)g(path)h(of)f(length)g Fu(d)c Fr(\025)h Fw(3.)94 1068 y(Then)f Fu(')p Fw(\()p Fu(S)6 b Fw(\))27 b(=)h(\()p Fu(a)770 1083 y Fq(k)807 1092 y Fo(1)857 1068 y Fw(+)12 b Fu(b)986 1083 y Fq(k)1023 1092 y Fo(1)1062 1068 y Fw(\))17 b Fr(\001)g(\001)g(\001)d Fw(\()p Fu(a)1338 1083 y Fq(k)1375 1095 y Fg(d)1427 1068 y Fw(+)e Fu(b)1556 1083 y Fq(k)1593 1095 y Fg(d)1633 1068 y Fw(\).)42 b(Expanding)28 b(this)f(pro)s(duct,)i(w)m(e)f(see)h(that)e(the)h(term)94 1184 y(that)f(con)m(tains)h(no)f(factors)g Fu(b)1161 1199 y Fq(k)1198 1209 y Fg(j)1262 1184 y Fw(is)g(equal)g(to)g Fu(S)6 b Fw(,)28 b(while)e(the)i(sum)f(of)g(the)g(terms)g(that)g(con)m (tain)g(ex-)94 1321 y(actly)f(one)g(factor)f Fu(b)814 1336 y Fq(k)851 1346 y Fg(j)914 1321 y Fw(is)g(easily)g(seen)j(to)d(b)s (e)h(cyclically)e(equiv)-5 b(alen)m(t)26 b(to)2696 1246 y Fm(P)2801 1272 y Fq(N)2801 1350 y(k)r Fp(=1)2951 1321 y Fu(b)2992 1336 y Fq(k)3035 1321 y Fu(@)3086 1336 y Fq(a)3123 1348 y Fg(k)3166 1321 y Fu(S)32 b Fw(\(cf.\()3434 1321 y SDict begin H.S end 3434 1321 a Fw(3.1)3558 1257 y SDict begin H.R end 3558 1257 a 3558 1321 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.3.1) cvn H.B /ANN pdfmark end 3558 1321 a Fw(\)\),)94 1458 y(and)27 b(the)g(rest)g(of)g(the)g (terms)f(are)h(cyclically)d(equiv)-5 b(alen)m(t)27 b(to)f(elemen)m(ts)h (of)2799 1383 y Fm(P)2904 1409 y Fq(N)2904 1487 y(k)r Fp(=1)3054 1458 y Fl(m)p Fw(\()p Fl(m)3244 1422 y Fq(d)p Fn(\000)p Fp(1)3385 1458 y Fr(\\)10 b Fu(I)e Fw(\))p Fu(b)3591 1473 y Fq(k)3634 1458 y Fw(.)194 1574 y(W)-8 b(riting)31 b(a)i(general)g(p)s(oten)m(tial)e Fu(S)j Fr(2)29 b Fl(m)1648 1538 y Fp(3)1721 1574 y Fw(as)34 b(a)e(linear)g(com)m(bination)f(of)i(cyclic)f(paths,w)m(e)j(see)94 1695 y(that)e Fu(')p Fw(\()p Fu(S)6 b Fw(\))22 b Fr(\000)h Fu(S)28 b Fr(\000)821 1621 y Fm(P)926 1647 y Fq(N)926 1725 y(k)r Fp(=1)1076 1695 y Fu(b)1117 1710 y Fq(k)1160 1695 y Fu(@)1211 1710 y Fq(a)1248 1722 y Fg(k)1291 1695 y Fu(S)39 b Fw(is)32 b(cyclically)f(equiv)-5 b(alen)m(t)32 b(to)2494 1621 y Fm(P)2599 1647 y Fq(N)2599 1725 y(k)r Fp(=1)2748 1695 y Fu(c)2790 1710 y Fq(k)2833 1695 y Fu(b)2874 1710 y Fq(k)2917 1695 y Fw(,)h(where)h(eac)m(h)g Fu(c)3521 1710 y Fq(k)3596 1695 y Fw(is)94 1812 y(of)f(the)g(form)1490 2001 y Fu(c)1532 2016 y Fq(k)1603 2001 y Fw(=)1747 1876 y Fq(N)1706 1906 y Fm(X)1719 2118 y Fq(`)p Fp(=1)1867 2001 y Fu(a)1918 2016 y Fq(`)2004 1876 y Fn(1)1968 1906 y Fm(X)1976 2118 y Fq(d)p Fp(=3)2128 2001 y Fu(c)2170 1950 y Fp(\()p Fq(d)p Fp(\))2170 2029 y Fq(k)r(`)94 2259 y Fw(with)44 b Fu(c)370 2208 y Fp(\()p Fq(d)p Fp(\))370 2287 y Fq(k)r(`)511 2259 y Fr(2)j Fl(m)700 2223 y Fq(d)p Fn(\000)p Fp(1)861 2259 y Fr(\\)30 b Fu(I)8 b Fw(.)76 b(Since)44 b Fu(I)51 b Fw(is)43 b(closed,)k(eac)m(h)d Fu(c)2181 2274 y Fq(k)2267 2259 y Fw(is)g(a)f(w)m(ell-de\014ned)h (elemen)m(t)f(of)g Fl(m)p Fu(I)8 b Fw(,)94 2375 y(implying)30 b(the)j(assertion)g(of)f(Lemma)1533 2375 y SDict begin H.S end 1533 2375 a Fw(4.11)1706 2311 y SDict begin H.R end 1706 2311 a 1706 2375 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.11) cvn H.B /ANN pdfmark end 1706 2375 a Fw(.)1851 b Fh(\003)194 2542 y Fw(W)-8 b(e)33 b(will)d(also)i(need)i (one)e(more)g(lemma)f(whose)i(pro)s(of)f(will)e(b)s(e)j(giv)m(en)g(in)f (Section)3276 2542 y SDict begin H.S end 3276 2542 a Fw(13)3374 2478 y SDict begin H.R end 3374 2478 a 3374 2542 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (section.13) cvn H.B /ANN pdfmark end 3374 2542 a Fw(.)94 2562 y SDict begin H.S end 94 2562 a 94 2562 a SDict begin 14 H.A end 94 2562 a 94 2562 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.4.12) cvn H.B /DEST pdfmark end 94 2562 a 155 x FA(Lemma)45 b(4.12.)g Fv(L)-5 b(et)41 b Fu(I)49 b Fv(b)-5 b(e)40 b(a)h(close)-5 b(d)39 b(ide)-5 b(al)40 b(of)h Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fv(,)g(and)f Fu(J)50 b Fv(b)-5 b(e)40 b(the)h(closur)-5 b(e)40 b(of)h(an)f(ide)-5 b(al)94 2833 y(gener)g(ate)g(d)44 b(by)h(\014nitely)f(many)g(elements)f Fu(f)1740 2848 y Fp(1)1780 2833 y Fu(;)17 b(f)1872 2848 y Fp(2)1911 2833 y Fu(;)g(:)g(:)g(:)f(;)h(f)2178 2848 y Fq(N)2245 2833 y Fv(,)47 b(which)c(ar)-5 b(e)44 b(bi-homo)-5 b(gene)g(ous)43 b(with)94 2950 y(r)-5 b(esp)g(e)g(ct)46 b(to)g(the)g(vertex)f(bigr)-5 b(ading.)76 b(Then)45 b(every)h(p)-5 b(otential)45 b(b)-5 b(elonging)44 b(to)i(the)g(ide)-5 b(al)45 b Fu(I)8 b(J)55 b Fv(is)94 3071 y(cyclic)-5 b(al)5 b(ly)35 b(e)-5 b(quivalent)34 b(to)h(an)g(element)f(of)g(the)h(form) 2096 2996 y Fm(P)2201 3023 y Fq(N)2201 3100 y(k)r Fp(=1)2350 3071 y Fu(b)2391 3086 y Fq(k)2434 3071 y Fu(f)2482 3086 y Fq(k)2525 3071 y Fv(,)g(wher)-5 b(e)34 b(al)5 b(l)35 b Fu(b)3046 3086 y Fq(k)3124 3071 y Fv(b)-5 b(elong)34 b(to)h Fu(I)8 b Fv(.)194 3246 y Fw(T)-8 b(o)33 b(pro)m(v)m(e)h(part)e (\(2\))g(of)g(Prop)s(osition)1605 3246 y SDict begin H.S end 1605 3246 a Fw(4.10)1778 3182 y SDict begin H.R end 1778 3182 a 1778 3246 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.10) cvn H.B /ANN pdfmark end 1778 3246 a Fw(,)h(w)m(e)h(construct)f(a)g(sequence)i(of)d Fu(N)10 b Fw(-tuples)1387 3404 y(\()p Fu(b)1466 3419 y Fp(1)p Fq(n)1548 3404 y Fu(;)17 b(:)g(:)g(:)f(;)h(b)1808 3419 y Fq(N)7 b(n)1918 3404 y Fw(\))98 b(\()p Fu(n)28 b Fr(\025)g Fw(1\))94 3562 y(of)j(elemen)m(ts)g(of)f Fl(m)786 3526 y Fp(2)857 3562 y Fw(and)h(the)g(corresp)s(onding)g(unitriangular)d (automorphisms)h Fu(')3171 3577 y Fq(n)3249 3562 y Fw(of)h Fu(R)q Fr(hh)p Fu(A)p Fr(ii)94 3678 y Fw(\(so)j(that)g Fu(')528 3693 y Fq(n)574 3678 y Fw(\()p Fu(a)663 3693 y Fq(k)706 3678 y Fw(\))28 b(=)f Fu(a)926 3693 y Fq(k)991 3678 y Fw(+)22 b Fu(b)1130 3693 y Fq(k)r(n)1249 3678 y Fw(for)32 b Fu(k)f Fw(=)c(1)p Fu(;)17 b(:)g(:)g(:)f(;)h(N)10 b Fw(\))32 b(suc)m(h)j(that,)d(for)g(all)e Fu(n)e Fr(\025)h Fw(1,)j(w)m(e)i(ha)m(v)m(e)94 3724 y SDict begin H.S end 94 3724 a 94 3724 a SDict begin 14 H.A end 94 3724 a 94 3724 a SDict begin [ /View [/XYZ H.V] /Dest (Item.7) cvn H.B /DEST pdfmark end 94 3724 a 243 3815 a Fw(\(1\))41 b Fu(b)450 3830 y Fq(k)r(n)564 3815 y Fr(2)28 b Fl(m)734 3779 y Fq(n)p Fp(+1)893 3815 y Fr(\\)23 b Fu(J)9 b Fw(\()p Fu(S)d Fw(\))32 b(for)g Fu(k)f Fw(=)d(1)p Fu(;)17 b(:)g(:)g(:)e(;)i(N) 10 b Fw(.)94 3840 y SDict begin H.S end 94 3840 a 94 3840 a SDict begin 14 H.A end 94 3840 a 94 3840 a SDict begin [ /View [/XYZ H.V] /Dest (Item.8) cvn H.B /DEST pdfmark end 94 3840 a 243 3942 a Fw(\(2\))41 b Fu(S)475 3906 y Fn(0)530 3942 y Fw(is)31 b(cyclically)f(equiv)-5 b(alen)m(t)32 b(to)f Fu(')1694 3957 y Fp(0)1733 3942 y Fu(')1797 3957 y Fp(1)1853 3942 y Fr(\001)17 b(\001)g(\001)e Fu(')2050 3957 y Fq(n)p Fn(\000)p Fp(1)2187 3942 y Fw(\()p Fu(S)26 b Fw(+)2408 3867 y Fm(P)2513 3894 y Fq(N)2513 3971 y(k)r Fp(=1)2662 3942 y Fu(b)2703 3957 y Fq(k)r(n)2789 3942 y Fu(@)2840 3957 y Fq(a)2877 3969 y Fg(k)2921 3942 y Fu(S)6 b Fw(\))31 b(\(with)g(the)i(con-)409 4058 y(v)m(en)m(tion)g (that)g Fu(')1029 4073 y Fp(0)1100 4058 y Fw(is)f(the)h(iden)m(tit)m(y) g(automorphism\).)194 4195 y(W)-8 b(e)47 b(pro)s(ceed)g(b)m(y)g (induction)e(on)i Fu(n)p Fw(.)85 b(In)46 b(the)h(basic)f(case)h Fu(n)52 b Fw(=)e(1,)g(the)d(existence)h(of)d(an)94 4312 y Fu(N)10 b Fw(-tuple)39 b(\()p Fu(b)550 4327 y Fp(11)625 4312 y Fu(;)17 b(:)g(:)g(:)f(;)h(b)885 4327 y Fq(N)7 b Fp(1)988 4312 y Fw(\))39 b(with)g(desired)g(prop)s(erties)g(follo)m (ws)f(from)g(Lemma)3014 4312 y SDict begin H.S end 3014 4312 a Fw(4.12)3188 4247 y SDict begin H.R end 3188 4247 a 3188 4312 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.12) cvn H.B /ANN pdfmark end 3188 4312 a 39 w Fw(applied)g(to)94 4428 y Fu(I)k Fw(=)34 b Fu(J)44 b Fw(=)34 b Fu(J)9 b Fw(\()p Fu(S)d Fw(\))36 b(and)g Fu(f)979 4443 y Fq(k)1056 4428 y Fw(=)e Fu(@)1217 4443 y Fq(a)1254 4455 y Fg(k)1297 4428 y Fu(S)42 b Fw(\(note)37 b(that)f Fu(J)9 b Fw(\()p Fu(S)d Fw(\))34 b Fr(\022)g Fl(m)2299 4392 y Fp(2)2339 4428 y Fw(,)k(since)e(\()p Fu(A;)17 b(S)6 b Fw(\))36 b(is)g(assumed)h(to)f(b)s(e)94 4544 y(reduced\).)194 4660 y(No)m(w)j(assume)g(that,)h(for)e(some)g Fu(n)g Fr(\025)g Fw(1,)i(w)m(e)f(ha)m(v)m(e)h(already)e(de\014ned)i(the)f (elemen)m(ts)g Fu(b)3435 4675 y Fq(k)r(`)3545 4660 y Fw(for)94 4777 y Fu(k)50 b Fw(=)c(1)p Fu(;)17 b(:)g(:)g(:)e(;)i(N)54 b Fw(and)43 b Fu(`)j Fw(=)g(1)p Fu(;)17 b(:)g(:)g(:)f(;)h(n)p Fw(,)46 b(satisfying)c(\(1\))h(and)h(\(2\).)75 b(Applying)43 b(Lemma)3358 4777 y SDict begin H.S end 3358 4777 a Fw(4.11)3531 4712 y SDict begin H.R end 3531 4712 a 3531 4777 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.11) cvn H.B /ANN pdfmark end 3531 4777 a 44 w Fw(to)94 4904 y Fu(b)135 4919 y Fq(k)209 4904 y Fw(=)29 b Fu(b)355 4919 y Fq(k)r(n)475 4904 y Fw(\(so)34 b(that)g Fu(')c Fw(=)g Fu(')1111 4919 y Fq(n)1158 4904 y Fw(\),)k(w)m(e)h(obtain)e(that)g Fu(')1983 4919 y Fq(n)2030 4904 y Fw(\()p Fu(S)6 b Fw(\))23 b Fr(\000)h Fw(\()p Fu(S)k Fw(+)2521 4829 y Fm(P)2627 4855 y Fq(N)2627 4933 y(k)r Fp(=1)2776 4904 y Fu(b)2817 4919 y Fq(k)r(n)2903 4904 y Fu(@)2954 4919 y Fq(a)2991 4931 y Fg(k)3034 4904 y Fu(S)6 b Fw(\))34 b(is)f(cyclically)94 5020 y(equiv)-5 b(alen)m(t)33 b(to)f(an)h(elemen)m(t)f(of)g Fl(m)p Fw(\()p Fl(m)1474 4984 y Fq(n)p Fp(+1)1634 5020 y Fr(\\)23 b Fu(J)9 b Fw(\()p Fu(S)d Fw(\)\))1966 4984 y Fp(2)2005 5020 y Fw(.)43 b(W)-8 b(e)33 b(ha)m(v)m(e)1033 5179 y Fl(m)p Fw(\()p Fl(m)1223 5138 y Fq(n)p Fp(+1)1383 5179 y Fr(\\)22 b Fu(J)9 b Fw(\()p Fu(S)d Fw(\)\))1714 5138 y Fp(2)1781 5179 y Fr(\022)28 b Fw(\()p Fl(m)2000 5138 y Fq(n)p Fp(+2)2160 5179 y Fr(\\)22 b Fu(J)9 b Fw(\()p Fu(S)d Fw(\)\))p Fu(J)j Fw(\()p Fu(S)d Fw(\))p Fu(:)94 5337 y Fw(This)40 b(implies)c(in)j(particular)e(that)i Fu(')1520 5352 y Fq(n)1567 5337 y Fw(\()p Fu(S)6 b Fw(\))26 b Fr(\000)h Fu(S)45 b Fw(is)39 b(cyclically)e(equiv)-5 b(alen)m(t)39 b(to)g(an)g(elemen)m(t)g(of)94 5453 y Fu(J)9 b Fw(\()p Fu(S)d Fw(\))299 5417 y Fp(2)339 5453 y Fw(.)86 b(Com)m(bining)45 b(Prop)s(osition)1502 5453 y SDict begin H.S end 1502 5453 a Fw(3.7)1626 5389 y SDict begin H.R end 1626 5389 a 1626 5453 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.3.7) cvn H.B /ANN pdfmark end 1626 5453 a 47 w Fw(with)i(the)g(already)g(pro)m(v)m(ed)h(part)f(\(1\))f(of)h (Prop)s(osi-)94 5569 y(tion)299 5569 y SDict begin H.S end 299 5569 a Fw(4.10)472 5505 y SDict begin H.R end 472 5505 a 472 5569 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.10) cvn H.B /ANN pdfmark end 472 5569 a Fw(,)37 b(w)m(e)g(conclude)g (that)f Fu(')1367 5584 y Fq(n)1414 5569 y Fw(\()p Fu(J)9 b Fw(\()p Fu(S)d Fw(\)\))33 b(=)h Fu(J)9 b Fw(\()p Fu(')2003 5584 y Fq(n)2050 5569 y Fw(\()p Fu(S)d Fw(\)\))33 b(=)h Fu(J)9 b Fw(\()p Fu(S)d Fw(\).)54 b(It)36 b(follo)m(ws)f(that)h Fu(')3371 5584 y Fq(n)3418 5569 y Fw(\()p Fu(S)6 b Fw(\))24 b Fr(\000)94 5696 y Fw(\()p Fu(S)f Fw(+)308 5621 y Fm(P)413 5648 y Fq(N)413 5725 y(k)r Fp(=1)563 5696 y Fu(b)604 5711 y Fq(k)r(n)690 5696 y Fu(@)741 5711 y Fq(a)778 5723 y Fg(k)821 5696 y Fu(S)6 b Fw(\))30 b(is)f(cyclically)g(equiv)-5 b(alen)m(t)30 b(to)f(an)h(elemen)m(t)g(of)g Fu(')2712 5711 y Fq(n)2759 5696 y Fw(\(\()p Fl(m)2911 5660 y Fq(n)p Fp(+2)3065 5696 y Fr(\\)18 b Fu(J)9 b Fw(\()p Fu(S)d Fw(\)\))p Fu(J)j Fw(\()p Fu(S)d Fw(\)\).)p eop end end %%Page: 18 18 TeXDict begin HPSdict begin 18 17 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.18) cvn H.B /DEST pdfmark end 94 218 a Fs(18)492 b(HARM)25 b(DERKSEN,)f(JERZY)i(WEYMAN,)f(AND)f(ANDREI)g(ZELEVINSKY)194 417 y Fw(Applying)33 b(Lemma)967 417 y SDict begin H.S end 967 417 a Fw(4.12)1140 353 y SDict begin H.R end 1140 353 a 1140 417 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.12) cvn H.B /ANN pdfmark end 1140 417 a 34 w Fw(to)g Fu(I)k Fw(=)29 b Fl(m)1555 381 y Fq(n)p Fp(+2)1715 417 y Fr(\\)24 b Fu(J)9 b Fw(\()p Fu(S)d Fw(\),)33 b Fu(J)39 b Fw(=)29 b Fu(J)9 b Fw(\()p Fu(S)d Fw(\))33 b(and)g Fu(f)2744 432 y Fq(k)2816 417 y Fw(=)d Fu(@)2973 432 y Fq(a)3010 444 y Fg(k)3053 417 y Fu(S)6 b Fw(,)33 b(w)m(e)i(see)g(that)94 533 y(ev)m(ery)30 b(elemen)m(t)d(of)g(\()p Fl(m)922 497 y Fq(n)p Fp(+2)1072 533 y Fr(\\)12 b Fu(J)d Fw(\()p Fu(S)d Fw(\)\))p Fu(J)j Fw(\()p Fu(S)d Fw(\))27 b(is)g(cyclically)f(equiv)-5 b(alen)m(t)27 b(to)h(an)f(elemen)m(t)h(of)f(the)h(form)1621 675 y Fq(N)1581 705 y Fm(X)1588 917 y Fq(k)r Fp(=1)1741 800 y Fu(b)1782 815 y Fq(k)r(;n)p Fp(+1)1978 800 y Fu(@)2029 815 y Fq(a)2066 827 y Fg(k)2109 800 y Fu(S)94 1083 y Fw(for)e(some)g(elemen)m(ts)h Fu(b)909 1098 y Fq(k)r(;n)p Fp(+1)1132 1083 y Fr(2)h Fl(m)1302 1047 y Fq(n)p Fp(+2)1449 1083 y Fr(\\)9 b Fu(J)g Fw(\()p Fu(S)d Fw(\).)41 b(It)26 b(follo)m(ws)f(that)h Fu(S)15 b Fw(+)2575 1008 y Fm(P)2680 1034 y Fq(N)2680 1112 y(k)r Fp(=1)2829 1083 y Fu(b)2870 1098 y Fq(k)r(n)2956 1083 y Fu(@)3007 1098 y Fq(a)3044 1110 y Fg(k)3088 1083 y Fu(S)32 b Fw(is)25 b(cyclically)94 1214 y(equiv)-5 b(alen)m(t)24 b(to)f Fu(')723 1229 y Fq(n)770 1214 y Fw(\()p Fu(S)10 b Fw(+)958 1139 y Fm(P)1062 1165 y Fq(N)1062 1243 y(k)r Fp(=1)1212 1214 y Fu(b)1253 1229 y Fq(k)r(;n)p Fp(+1)1448 1214 y Fu(@)1499 1229 y Fq(a)1536 1241 y Fg(k)1579 1214 y Fu(S)c Fw(\).)41 b(Th)m(us,)27 b(conditions)22 b(\(1\))h(and)h(\(2\))f(get)h(satis\014ed)g(with)94 1330 y Fu(n)33 b Fw(replaced)g(b)m(y)g Fu(n)23 b Fw(+)f(1,)32 b(completing)f(our)h(inductiv)m(e)h(step.)194 1446 y(In)44 b(view)g(of)g(condition)e(\(1\),)47 b(lim)1459 1461 y Fq(n)p Fn(!1)1664 1446 y Fu(')1728 1461 y Fp(1)1784 1446 y Fr(\001)17 b(\001)g(\001)e Fu(')1981 1461 y Fq(n)2071 1446 y Fw(is)44 b(a)f(w)m(ell-de\014ned)h(automorphism)e Fu(')i Fw(of)94 1563 y Fu(R)q Fr(hh)p Fu(A)p Fr(ii)34 b Fw(suc)m(h)i(that)e Fu(')p Fw(\()p Fu(u)p Fw(\))23 b Fr(\000)h Fu(u)31 b Fr(2)g Fu(J)9 b Fw(\()p Fu(S)d Fw(\))34 b(for)g(all)f Fu(u)d Fr(2)i Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fw(.)48 b(P)m(assing)35 b(to)f(the)h(limit)c Fu(n)g Fr(!)g(1)94 1679 y Fw(in)e(condition)f(\(2\),)i(w)m(e)g (conclude)g(that)g Fu(S)1625 1643 y Fn(0)1677 1679 y Fw(is)f(cyclically)e(equiv)-5 b(alen)m(t)30 b(to)f Fu(')p Fw(\()p Fu(S)6 b Fw(\),)29 b(completing)f(the)94 1795 y(pro)s(of)k(of)g(part)h(\(2\))f(of)g(Prop)s(osition)1465 1795 y SDict begin H.S end 1465 1795 a Fw(4.10)1638 1731 y SDict begin H.R end 1638 1731 a 1638 1795 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.10) cvn H.B /ANN pdfmark end 1638 1795 a Fw(.)1919 b Fh(\003)94 1977 y Fv(Pr)-5 b(o)g(of)35 b(of)g(Pr)-5 b(op)g(osition)989 1996 y SDict begin H.S end 989 1996 a -19 x Fv(4.9)1118 1912 y SDict begin H.R end 1118 1912 a 1118 1977 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.9) cvn H.B /ANN pdfmark end 1118 1977 a Fv(.)42 b Fw(W)-8 b(e)40 b(abbreviate)f Fu(J)49 b Fw(=)39 b Fu(J)9 b Fw(\()p Fu(S)d Fw(\))40 b(and)f Fu(J)2573 1941 y Fn(0)2636 1977 y Fw(=)h Fu(J)9 b Fw(\()p Fu(S)2919 1941 y Fn(0)2942 1977 y Fw(\))39 b(\(understo)s(o)s(d)h(as)94 2093 y(the)34 b(Jacobian)f(ideals)g(of)g Fu(S)39 b Fw(and)33 b Fu(S)1412 2057 y Fn(0)1469 2093 y Fw(in)g Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fw(\).)45 b(As)34 b(in)f(Prop)s(osition)2784 2093 y SDict begin H.S end 2784 2093 a Fw(4.5)2908 2029 y SDict begin H.R end 2908 2029 a 2908 2093 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.5) cvn H.B /ANN pdfmark end 2908 2093 a Fw(,)h(let)f Fu(L)h Fw(denote)g(the)94 2209 y(closure)f(of)f(the)h(t)m(w)m(o-sided)g (ideal)e(in)h Fu(R)q Fr(hh)p Fu(A)22 b Fr(\010)h Fu(C)7 b Fr(ii)32 b Fw(generated)h(b)m(y)h Fu(C)7 b Fw(.)43 b(Then)34 b(w)m(e)f(ha)m(v)m(e)94 2379 y(\(4.10\))484 2379 y SDict begin H.S end 484 2379 a 484 2379 a SDict begin 14 H.A end 484 2379 a 484 2379 a SDict begin [ /View [/XYZ H.V] /Dest (equation.4.10) cvn H.B /DEST pdfmark end 484 2379 a Fu(R)q Fr(hh)p Fu(A)22 b Fr(\010)h Fu(C)7 b Fr(ii)27 b Fw(=)g Fu(R)q Fr(hh)p Fu(A)p Fr(ii)22 b(\010)g Fu(L;)115 b(J)9 b Fw(\()p Fu(S)28 b Fw(+)22 b Fu(T)14 b Fw(\))27 b(=)h Fu(J)j Fr(\010)23 b Fu(L;)114 b(J)9 b Fw(\()p Fu(S)2836 2338 y Fn(0)2882 2379 y Fw(+)22 b Fu(T)14 b Fw(\))27 b(=)g Fu(J)3282 2338 y Fn(0)3328 2379 y Fr(\010)c Fu(L:)194 2548 y Fw(Let)g Fu(')g Fw(b)s(e)h(an)f Fu(R)q Fw(-algebra)f (automorphism)f(of)h Fu(R)q Fr(hh)p Fu(A)s Fr(\010)s Fu(C)7 b Fr(ii)p Fw(,)25 b(suc)m(h)f(that)f Fu(')p Fw(\()p Fu(S)9 b Fw(+)s Fu(T)14 b Fw(\))23 b(is)f(cyclically)94 2665 y(equiv)-5 b(alen)m(t)33 b(to)f Fu(S)743 2628 y Fn(0)788 2665 y Fw(+)22 b Fu(T)14 b Fw(.)43 b(In)33 b(view)g(of)f(\() 1523 2665 y SDict begin H.S end 1523 2665 a Fw(4.10)1697 2600 y SDict begin H.R end 1697 2600 a 1697 2665 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.4.10) cvn H.B /ANN pdfmark end 1697 2665 a Fw(\))g(and)h(Prop)s(osition)2482 2665 y SDict begin H.S end 2482 2665 a Fw(3.7)2606 2600 y SDict begin H.R end 2606 2600 a 2606 2665 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.3.7) cvn H.B /ANN pdfmark end 2606 2665 a Fw(,)g(w)m(e)h(ha)m(v)m(e)94 2834 y(\(4.11\))1466 2834 y SDict begin H.S end 1466 2834 a 1466 2834 a SDict begin 14 H.A end 1466 2834 a 1466 2834 a SDict begin [ /View [/XYZ H.V] /Dest (equation.4.11) cvn H.B /DEST pdfmark end 1466 2834 a Fu(')p Fw(\()p Fu(J)d Fr(\010)23 b Fu(L)p Fw(\))28 b(=)f Fu(J)2051 2793 y Fn(0)2097 2834 y Fr(\010)22 b Fu(L:)194 3004 y Fw(Let)43 b Fu( )48 b Fw(:)c Fu(R)q Fr(hh)p Fu(A)p Fr(ii)f(!)h Fu(R)q Fr(hh)p Fu(A)p Fr(ii)e Fw(denote)h(the)g(restriction)e(to)h Fu(R)q Fr(hh)p Fu(A)p Fr(ii)f Fw(of)h(the)h(comp)s(osition)94 3120 y Fu(p')p Fw(,)53 b(where)c Fu(p)g Fw(is)f(the)g(pro)5 b(jection)49 b(of)f Fu(R)q Fr(hh)p Fu(A)32 b Fr(\010)i Fu(C)7 b Fr(ii)47 b Fw(on)m(to)i Fu(R)q Fr(hh)p Fu(A)p Fr(ii)e Fw(along)g Fu(L)p Fw(.)91 b(In)49 b(view)g(of)94 3236 y(Prop)s(osition)619 3236 y SDict begin H.S end 619 3236 a Fw(4.10)793 3172 y SDict begin H.R end 793 3172 a 793 3236 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.10) cvn H.B /ANN pdfmark end 793 3236 a Fw(,)32 b(it)g(su\016ces)j(to)d (sho)m(w)i(the)f(follo)m(wing:)577 3406 y SDict begin H.S end 577 3406 a 577 3406 a SDict begin 14 H.A end 577 3406 a 577 3406 a SDict begin [ /View [/XYZ H.V] /Dest (equation.4.12) cvn H.B /DEST pdfmark end 577 3406 a 164 w Fu( )k Fw(is)32 b(an)g Fu(R)q Fw(-algebra)f(automorphism)g(of)h Fu(R)q Fr(hh)p Fu(A)p Fr(ii)g Fw(suc)m(h)i(that)-2921 b(\(4.12\))660 3551 y Fu(S)726 3515 y Fn(0)772 3551 y Fr(\000)22 b Fu( )t Fw(\()p Fu(S)6 b Fw(\))33 b(is)f(cyclically)e (equiv)-5 b(alen)m(t)33 b(to)f(an)g(elemen)m(t)h(of)f Fu( )t Fw(\()p Fu(J)2991 3515 y Fp(2)3030 3551 y Fw(\).)94 3721 y(Clearly)-8 b(,)40 b Fu( )j Fw(is)38 b(a)h(homomorphism)d(of)j Fu(R)q Fw(-algebras,)g(and)g(so)g(can)g(b)s(e)h(represen)m(ted)h(b)m(y) f(a)e(pair)94 3837 y(\()p Fu( )199 3801 y Fp(\(1\))294 3837 y Fu(;)17 b( )405 3801 y Fp(\(2\))499 3837 y Fw(\))34 b(as)g(in)f(Prop)s(osition)1334 3837 y SDict begin H.S end 1334 3837 a Fw(2.5)1458 3773 y SDict begin H.R end 1458 3773 a 1458 3837 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.2.5) cvn H.B /ANN pdfmark end 1458 3837 a Fw(.)48 b(T)-8 b(o)34 b(sho)m(w)i(that)e Fu( )k Fw(is)33 b(an)h(automorphism)e(of)i Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fw(,)f(it)94 3956 y(su\016ces)39 b(to)c(sho)m(w)j(that)e Fu( )1080 3920 y Fp(\(1\))1210 3956 y Fw(is)g(an)g Fu(R)q Fw(-bimo)s(dule)d(automorphism)h(of)i Fu(A)p Fw(.)54 b(By)37 b(the)f(de\014nition,)94 4073 y(if)c(w)m(e)i(write)e(the)h Fu(R)q Fw(-bimo)s(dule)d(automorphism)g Fu(')1987 4036 y Fp(\(1\))2114 4073 y Fw(of)i Fu(A)22 b Fr(\010)h Fu(C)39 b Fw(as)33 b(a)g(matrix)1564 4158 y Fm(\022)1639 4238 y Fu(')1703 4253 y Fq(AA)1898 4238 y Fu(')1962 4253 y Fq(AC)1638 4354 y Fu(')1702 4369 y Fq(C)5 b(A)1897 4354 y Fu(')1961 4369 y Fq(C)g(C)2075 4158 y Fm(\023)2165 4298 y Fu(;)94 4540 y Fw(then)34 b Fu( )384 4504 y Fp(\(1\))506 4540 y Fw(=)27 b Fu(')673 4555 y Fq(AA)783 4540 y Fw(.)43 b(Since)998 4710 y Fu(')p Fw(\()p Fu(C)7 b Fw(\))28 b Fr(\032)g Fu(')p Fw(\()p Fu(J)j Fr(\010)23 b Fu(L)p Fw(\))28 b(=)f Fu(J)1933 4669 y Fn(0)1979 4710 y Fr(\010)22 b Fu(L)28 b Fr(\022)h Fl(m)p Fw(\()p Fu(A)p Fw(\))2503 4669 y Fp(2)2565 4710 y Fr(\010)22 b Fu(L;)94 4883 y Fw(it)34 b(follo)m(ws)f(that)h Fu(')793 4898 y Fq(AC)935 4883 y Fw(=)d(0,)j(implying)e(that)i Fu( )1838 4846 y Fp(\(1\))1962 4883 y Fw(=)d Fu(')2133 4898 y Fq(AA)2277 4883 y Fw(is)j(an)g Fu(R)q Fw(-bimo)s(dule)d (automorphism)94 4999 y(of)i Fu(A)p Fw(,)f(and)h(that)f Fu( )37 b Fw(is)32 b(an)g(automorphism)f(of)h Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fw(.)194 5115 y(Since)23 b Fu(S)505 5079 y Fn(0)531 5115 y Fw(+)s Fu(T)36 b Fw(is)23 b(cyclically)e(equiv) -5 b(alen)m(t)23 b(to)g Fu(')p Fw(\()p Fu(S)9 b Fw(+)s Fu(T)14 b Fw(\),)23 b(it)f(follo)m(ws)g(that)h Fu(S)2845 5079 y Fn(0)2871 5115 y Fr(\000)s Fu( )t Fw(\()p Fu(S)6 b Fw(\))23 b(is)f(cyclically)94 5231 y(equiv)-5 b(alen)m(t)32 b(to)f Fu(p')p Fw(\()p Fu(T)14 b Fw(\).)43 b(Since)32 b Fu(T)41 b Fr(2)28 b Fu(C)1528 5195 y Fp(2)1568 5231 y Fw(,)k(the)g(claim)d(that)j Fu(S)2330 5195 y Fn(0)2373 5231 y Fr(\000)21 b Fu( )t Fw(\()p Fu(S)6 b Fw(\))31 b(is)h(cyclically)e(equiv)-5 b(alen)m(t)94 5347 y(to)40 b(an)f(elemen)m(t)h(of)f Fu( )t Fw(\()p Fu(J)1017 5311 y Fp(2)1056 5347 y Fw(\))h(follo)m(ws)e(from)h(the)h(fact)f(that)h Fu(p')p Fw(\()p Fu(L)p Fw(\))f Fr(\022)h Fu( )t Fw(\()p Fu(J)9 b Fw(\),)42 b(or,)f(equiv)-5 b(alen)m(tly)d(,)94 5464 y(that)41 b Fu(')p Fw(\()p Fu(L)p Fw(\))f Fr(\022)i Fu(')p Fw(\()p Fu(J)9 b Fw(\))27 b(+)g Fu(L)p Fw(.)67 b(Applying)40 b(the)g(in)m(v)m(erse)i(automorphism)c Fu(')2826 5428 y Fn(\000)p Fp(1)2961 5464 y Fw(to)i(b)s(oth)g(sides,)i (it)94 5580 y(su\016ces)37 b(to)d(sho)m(w)h(that)f Fu(L)d Fr(\022)g Fu(J)h Fw(+)23 b Fu(')1459 5544 y Fn(\000)p Fp(1)1554 5580 y Fw(\()p Fu(L)p Fw(\).)48 b(Using)34 b(the)h(ob)m(vious)f(symmetry)h(b)s(et)m(w)m(een)h Fu(J)43 b Fw(and)94 5696 y Fu(J)157 5660 y Fn(0)181 5696 y Fw(,)33 b(it)e(is)h(enough)h(to)f(sho)m(w)i(the)f(inclusion)e Fu(L)d Fr(\022)g Fu(J)1970 5660 y Fn(0)2016 5696 y Fw(+)22 b Fu(')p Fw(\()p Fu(L)p Fw(\).)p eop end end %%Page: 19 19 TeXDict begin HPSdict begin 19 18 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.19) cvn H.B /DEST pdfmark end 94 218 a 1179 w Fs(QUIVERS)31 b(WITH)i(POTENTIALS)g(I)1102 b(19)194 417 y Fw(Let)34 b(us)h(abbreviate)e Fu(M)41 b Fw(=)30 b Fl(m)p Fw(\()p Fu(A)23 b Fr(\010)h Fu(C)7 b Fw(\),)34 b(and)g Fu(I)j Fw(=)30 b Fu(J)2145 381 y Fn(0)2192 417 y Fw(+)22 b Fu(')p Fw(\()p Fu(L)p Fw(\).)48 b(Since)34 b Fu(')p Fw(\()p Fu(J)9 b Fw(\))29 b Fr(\022)i Fu(J)3230 381 y Fn(0)3276 417 y Fr(\010)24 b Fu(L)p Fw(,)34 b(and)94 533 y Fu(J)39 b Fr(\022)30 b Fl(m)p Fw(\()p Fu(A)p Fw(\))519 497 y Fp(2)559 533 y Fw(,)k(it)e(follo)m(ws)h(that)g Fu(')p Fw(\()p Fu(J)9 b Fw(\))30 b Fr(\022)g Fu(J)1655 497 y Fn(0)1701 533 y Fr(\010)23 b Fw(\()p Fu(L)g Fr(\\)h Fu(M)2122 497 y Fp(2)2162 533 y Fw(\))29 b(=)h Fu(J)2398 497 y Fn(0)2444 533 y Fw(+)23 b Fu(M)10 b(L)24 b Fw(+)e Fu(LM)10 b Fw(.)48 b(Therefore,)35 b(w)m(e)94 649 y(ha)m(v)m(e)918 766 y Fu(L)28 b Fr(\022)g Fu(J)1180 725 y Fn(0)1225 766 y Fw(+)22 b Fu(L)29 b Fw(=)e Fu(')p Fw(\()p Fu(J)9 b Fw(\))22 b(+)g Fu(')p Fw(\()p Fu(L)p Fw(\))28 b Fr(\022)g Fu(I)i Fw(+)22 b Fu(M)10 b(L)23 b Fw(+)f Fu(LM)5 b(:)94 902 y Fw(Substituting)28 b(this)g(upp)s(er)h(b)s(ound)g(for)f Fu(L)g Fw(in)m(to)g(its)g(righ)m(t)f(hand)i(side,)h(w)m(e)f(deduce)h (the)f(inclusion)1211 1059 y Fu(L)f Fr(\022)g Fu(I)i Fw(+)22 b Fu(M)1685 1018 y Fp(2)1725 1059 y Fu(L)h Fw(+)f Fu(M)10 b(LM)34 b Fw(+)22 b Fu(LM)2478 1018 y Fp(2)2518 1059 y Fu(:)94 1216 y Fw(Con)m(tin)m(uing)32 b(in)g(the)h(same)g(w)m(a) m(y)-8 b(,)33 b(for)g(ev)m(ery)h Fu(n)28 b(>)f Fw(0,)33 b(w)m(e)g(ha)m(v)m(e)h(the)f(inclusion)1097 1448 y Fu(L)28 b Fr(\022)g Fu(I)i Fw(+)1517 1323 y Fq(n)1467 1353 y Fm(X)1474 1565 y Fq(k)r Fp(=0)1627 1448 y Fu(M)1731 1407 y Fq(k)1775 1448 y Fu(LM)1945 1407 y Fq(n)p Fn(\000)p Fq(k)2114 1448 y Fr(\022)e Fu(I)i Fw(+)22 b Fu(M)2494 1407 y Fq(n)p Fp(+1)2632 1448 y Fu(:)94 1712 y Fw(In)35 b(view)f(of)f(\()594 1712 y SDict begin H.S end 594 1712 a Fw(2.3)719 1648 y SDict begin H.R end 719 1648 a 719 1712 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.2.3) cvn H.B /ANN pdfmark end 719 1712 a Fw(\),)h(it)f(follo)m(ws)g(that)h Fu(L)g Fw(is)f(con)m(tained)h(in)p 2210 1632 51 4 v 34 w Fu(I)7 b Fw(,)35 b(the)f(closure)g(of)g Fu(I)41 b Fw(in)34 b Fu(R)q Fr(hh)p Fu(A)22 b Fr(\010)i Fu(C)7 b Fr(ii)p Fw(.)94 1829 y(Ho)m(w)m(ev)m(er,)40 b(it)c(is)g(easy)i(to)e(see)h(that) g Fu(I)42 b Fw(=)34 b Fu(J)1700 1792 y Fn(0)1748 1829 y Fw(+)25 b Fu(')p Fw(\()p Fu(L)p Fw(\))37 b(is)f(closed)h(in)e Fu(R)q Fr(hh)p Fu(A)25 b Fr(\010)g Fu(C)7 b Fr(ii)36 b Fw(\(indeed,)i(the)94 1945 y(closedness)29 b(of)e Fu(I)35 b Fw(is)27 b(equiv)-5 b(alen)m(t)27 b(to)g(that)g(of)g Fu(')1776 1909 y Fn(\000)p Fp(1)1870 1945 y Fw(\()p Fu(I)8 b Fw(\))27 b(=)h Fu(')2192 1909 y Fn(\000)p Fp(1)2286 1945 y Fw(\()p Fu(J)2387 1909 y Fn(0)2411 1945 y Fw(\))11 b(+)g Fu(L)p Fw(,)29 b(and)e(so,)i(b)m(y)f(symmetry)-8 b(,)28 b(it)94 2061 y(is)d(enough)g(to)g(sho)m(w)h(that)f Fu(')p Fw(\()p Fu(J)9 b Fw(\))e(+)g Fu(L)24 b Fw(is)g(closed;)k(but)d (this)g(is)f(clear)g(since)i Fu(')p Fw(\()p Fu(J)9 b Fw(\))e(+)g Fu(L)27 b Fw(=)g Fu(p)3285 2025 y Fn(\000)p Fp(1)3380 2061 y Fw(\()p Fu( )t Fw(\()p Fu(J)9 b Fw(\)\))94 2177 y(is)35 b(the)g(in)m(v)m(erse)h(image)d(of)h(the)h(closed)g(ideal) e Fu( )t Fw(\()p Fu(J)9 b Fw(\))34 b(of)h Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fw(\).)48 b(This)35 b(completes)g(the)g(pro)s(ofs)94 2293 y(of)e(Prop)s(osition)730 2293 y SDict begin H.S end 730 2293 a Fw(4.9)855 2229 y SDict begin H.R end 855 2229 a 855 2293 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.9) cvn H.B /ANN pdfmark end 855 2293 a 33 w Fw(and)f(Theorem)1489 2293 y SDict begin H.S end 1489 2293 a Fw(4.6)1614 2229 y SDict begin H.R end 1614 2229 a 1614 2293 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.6) cvn H.B /ANN pdfmark end 1614 2293 a Fw(.)1943 b Fh(\003)94 2363 y SDict begin H.S end 94 2363 a 94 2363 a SDict begin 14 H.A end 94 2363 a 94 2363 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.4.13) cvn H.B /DEST pdfmark end 94 2363 a 105 x FA(De\014nition)54 b(4.13.)49 b Fw(W)-8 b(e)48 b(call)e(the)i(comp)s(onen)m(t)f(\()p Fu(A)2098 2483 y Fp(red)2200 2468 y Fu(;)17 b(S)2304 2483 y Fp(red)2406 2468 y Fw(\))48 b(in)e(the)i(decomp)s(osition)e(\() 3499 2468 y SDict begin H.S end 3499 2468 a Fw(4.5)3623 2404 y SDict begin H.R end 3623 2404 a 3623 2468 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.4.5) cvn H.B /ANN pdfmark end 3623 2468 a Fw(\))94 2584 y(the)37 b Fv(r)-5 b(e)g(duc)g(e)g(d)38 b(p)-5 b(art)46 b Fw(of)36 b(a)g(QP)g(\()p Fu(A;)17 b(S)6 b Fw(\))36 b(\(b)m(y)h(Theorem)2091 2584 y SDict begin H.S end 2091 2584 a Fw(4.6)2216 2520 y SDict begin H.R end 2216 2520 a 2216 2584 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.6) cvn H.B /ANN pdfmark end 2216 2584 a Fw(,)g(it)e(is)h(determined)g(b)m(y)h(\()p Fu(A;)17 b(S)6 b Fw(\))36 b(up)h(to)94 2700 y(righ)m(t-equiv)-5 b(alence\).)94 2783 y SDict begin H.S end 94 2783 a 94 2783 a SDict begin 14 H.A end 94 2783 a 94 2783 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.4.14) cvn H.B /DEST pdfmark end 94 2783 a 92 x FA(De\014nition)33 b(4.14.)39 b Fw(W)-8 b(e)30 b(call)e(a)h(quiv)m(er)h Fu(Q)f Fw(\(as)h(w)m(ell)e(as)i (its)f(arro)m(w)g(span)h Fu(A)p Fw(\))f(2)p Fv(-acyclic)34 b Fw(if)29 b(it)f(has)94 2991 y(no)33 b(orien)m(ted)g(2-cycles,)g (i.e.,)f(satis\014es)h(the)g(follo)m(wing)d(condition:)94 3148 y(\(4.13\))692 3148 y SDict begin H.S end 692 3148 a 692 3148 a SDict begin 14 H.A end 692 3148 a 692 3148 a SDict begin [ /View [/XYZ H.V] /Dest (equation.4.13) cvn H.B /DEST pdfmark end 692 3148 a Fw(F)-8 b(or)32 b(ev)m(ery)i(v)m(ertices)g Fu(i)28 b Fr(6)p Fw(=)f Fu(j)6 b Fw(,)33 b(either)f Fu(A)2096 3163 y Fq(i;j)2204 3148 y Fw(=)c Fr(f)p Fw(0)p Fr(g)k Fw(or)g Fu(A)2681 3163 y Fq(j;i)2785 3148 y Fw(=)c Fr(f)p Fw(0)p Fr(g)p Fw(.)194 3322 y(In)39 b(the)g(rest)h(of)e(this)g(section) h(w)m(e)h(study)g(the)f(conditions)f(on)g(a)h(QP)g(\()p Fu(A;)17 b(S)6 b Fw(\))38 b(guaran)m(teeing)94 3438 y(that)33 b(its)f(reduced)i(part)e(is)g(2-acyclic.)43 b(W)-8 b(e)33 b(need)g(some)g(preparation.)194 3555 y(F)-8 b(or)44 b(a)h(quiv)m(er)g Fu(Q)h Fw(with)e(the)h(arro)m(w)g(span)h Fu(A)p Fw(,)i(let)c Fr(C)55 b Fw(=)49 b Fr(C)6 b Fw(\()p Fu(A)p Fw(\))45 b(denote)g(the)h(set)f(of)g(cyclic)94 3671 y(paths)32 b(on)f Fu(A)g Fw(up)g(to)f(cyclical)g(equiv)-5 b(alence.)43 b(Th)m(us,)33 b Fr(C)k Fw(is)30 b(either)h(empt)m(y)g (\(if)f Fu(Q)h Fw(has)g(no)g(orien)m(ted)94 3787 y(cycles)45 b(at)e(all\),)g(or)g(coun)m(table.)75 b(The)45 b(space)f(of)e(p)s(oten) m(tials)g(up)i(to)f(cyclical)e(equiv)-5 b(alence)44 b(is)94 3903 y(naturally)31 b(iden)m(ti\014ed)h(with)g Fu(K)1251 3867 y Fn(C)1296 3903 y Fw(.)44 b(W)-8 b(e)32 b(sa)m(y)i(that)e(a)g Fu(K)7 b Fw(-v)-5 b(alued)31 b(function)h(on)g Fu(K)3030 3867 y Fn(C)3108 3903 y Fw(is)g Fv(p)-5 b(olynomial)94 4020 y Fw(if)27 b(it)g(dep)s(ends)j(on)e(\014nitely)f(man)m(y)h(comp)s (onen)m(ts)h(of)e(a)h(p)s(oten)m(tial)e Fu(S)34 b Fw(and)28 b(can)g(b)s(e)h(expressed)h(as)f(a)94 4136 y(p)s(olynomial)22 b(in)j(these)i(comp)s(onen)m(ts.)42 b(F)-8 b(or)24 b(a)i(nonzero)g(p)s (olynomial)c(function)j Fu(F)14 b Fw(,)27 b(w)m(e)f(denote)g(b)m(y)94 4252 y Fu(U)10 b Fw(\()p Fu(F)k Fw(\))28 b Fr(\032)h Fu(K)547 4216 y Fn(C)622 4252 y Fw(the)i(set)g(of)f(all)e(p)s(oten)m (tials)i Fu(S)36 b Fw(suc)m(h)c(that)e Fu(F)14 b Fw(\()p Fu(S)6 b Fw(\))27 b Fr(6)p Fw(=)g(0.)43 b(By)31 b(a)f(regular)g (function)f(on)94 4368 y Fu(U)10 b Fw(\()p Fu(F)k Fw(\))28 b(w)m(e)h(mean)e(a)g(ratio)f(of)h(t)m(w)m(o)h(p)s(olynomial)c (functions)j(on)g Fu(K)2468 4332 y Fn(C)2541 4368 y Fw(suc)m(h)i(that)e (the)h(denominator)94 4485 y(v)-5 b(anishes)37 b(no)m(where)g(on)e Fu(U)10 b Fw(\()p Fu(F)k Fw(\);)38 b(in)d(particular,)g(an)m(y)h (function)f(of)g(the)h(form)f Fu(G=F)3190 4448 y Fq(n)3236 4485 y Fw(,)i(where)f Fu(G)94 4601 y Fw(is)30 b(a)g(p)s(olynomial,)d (is)j(regular)f(on)i Fu(U)10 b Fw(\()p Fu(F)k Fw(\).)43 b(If)30 b Fu(A)1823 4565 y Fn(0)1876 4601 y Fw(is)g(the)h(arro)m(w)f (span)h(of)f(another)g(quiv)m(er)h Fu(Q)3469 4565 y Fn(0)3493 4601 y Fw(,)f(w)m(e)94 4721 y(sa)m(y)h(that)f(a)f(map)g Fu(K)851 4685 y Fn(C)t Fp(\()p Fq(A)p Fp(\))1032 4721 y Fr(!)e Fu(K)1249 4685 y Fn(C)t Fp(\()p Fq(A)1370 4662 y Ff(0)1393 4685 y Fp(\))1454 4721 y Fw(is)j Fv(p)-5 b(olynomial)39 b Fw(if)28 b(its)i(ev)m(ery)h(comp)s(onen)m(t)f(is)f(a)h (p)s(olynomial)94 4837 y(function;)51 b(similarly)-8 b(,)45 b(a)f(map)h Fu(U)10 b Fw(\()p Fu(F)k Fw(\))49 b Fr(!)f Fu(K)1795 4801 y Fn(C)t Fp(\()p Fq(A)1916 4778 y Ff(0)1939 4801 y Fp(\))2016 4837 y Fw(is)c Fv(r)-5 b(e)g(gular)56 b Fw(if)44 b(its)g(ev)m(ery)j(comp)s(onen)m(t)e(is)g(a) 94 4954 y(regular)32 b(function)g(on)h Fu(U)10 b Fw(\()p Fu(F)k Fw(\).)194 5070 y(No)m(w)34 b(supp)s(ose)g(that)f(the)g(arro)m (w)g(span)h Fu(A)f Fw(satis\014es)g(\()2170 5070 y SDict begin H.S end 2170 5070 a Fw(4.1)2295 5006 y SDict begin H.R end 2295 5006 a 2295 5070 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.4.1) cvn H.B /ANN pdfmark end 2295 5070 a Fw(\),)g(and)g(let)f Fr(f)p Fu(a)2825 5085 y Fp(1)2864 5070 y Fu(;)17 b(b)2949 5085 y Fp(1)2989 5070 y Fu(;)g(:)g(:)g(:)f(;)h (a)3259 5085 y Fq(N)3326 5070 y Fu(;)g(b)3411 5085 y Fq(N)3478 5070 y Fr(g)33 b Fw(b)s(e)94 5186 y(an)m(y)j(maximal)c (collection)h(of)h(distinct)h(arro)m(ws)g(in)f Fu(Q)i Fw(suc)m(h)g(that)f Fu(b)2607 5201 y Fq(k)2650 5186 y Fu(a)2701 5201 y Fq(k)2779 5186 y Fw(is)f(a)h(cyclic)g(2-path)f(for)94 5302 y Fu(k)d Fw(=)d(1)p Fu(;)17 b(:)g(:)g(:)e(;)i(N)10 b Fw(.)42 b(Then)27 b(the)f(quiv)m(er)h(obtained)e(from)f Fu(Q)i Fw(b)m(y)h(remo)m(ving)e(this)h(collection)d(of)j(arro)m(ws)94 5419 y(is)32 b(clearly)f(2-acyclic.)42 b(T)-8 b(o)31 b(suc)m(h)j(a)d(collection)f(w)m(e)j(asso)s(ciate)e(a)g(nonzero)i(p)s (olynomial)28 b(function)94 5535 y(on)33 b Fu(K)320 5499 y Fn(C)t Fp(\()p Fq(A)p Fp(\))505 5535 y Fw(giv)m(en)g(b)m(y)94 5692 y(\(4.14\))1176 5692 y SDict begin H.S end 1176 5692 a 1176 5692 a SDict begin 14 H.A end 1176 5692 a 1176 5692 a SDict begin [ /View [/XYZ H.V] /Dest (equation.4.14) cvn H.B /DEST pdfmark end 1176 5692 a Fu(D)1260 5651 y Fq(b)1290 5660 y Fo(1)1324 5651 y Fq(;:::)o(;b)1453 5662 y Fg(N)1257 5716 y Fq(a)1294 5725 y Fo(1)1329 5716 y Fq(;:::)n(;a)1464 5727 y Fg(N)1526 5692 y Fw(\()p Fu(S)6 b Fw(\))28 b(=)f(det)q(\()p Fu(x)2028 5707 y Fq(b)2058 5715 y Fg(q)2093 5707 y Fq(a)2130 5715 y Fg(p)2170 5692 y Fw(\))2208 5707 y Fq(p;q)r Fp(=1)p Fq(;:::)n(;N)2553 5692 y Fu(;)p eop end end %%Page: 20 20 TeXDict begin HPSdict begin 20 19 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.20) cvn H.B /DEST pdfmark end 94 218 a Fs(20)492 b(HARM)25 b(DERKSEN,)f(JERZY)i(WEYMAN,)f(AND)f(ANDREI)g(ZELEVINSKY)94 417 y Fw(where)35 b Fu(x)432 432 y Fq(b)462 440 y Fg(q)497 432 y Fq(a)534 440 y Fg(p)608 417 y Fw(is)e(the)h(sum)f(of)g(the)g(co)s (e\016cien)m(ts)i(of)e Fu(b)2011 432 y Fq(q)2049 417 y Fu(a)2100 432 y Fq(p)2173 417 y Fw(and)h Fu(a)2415 432 y Fq(p)2455 417 y Fu(b)2496 432 y Fq(q)2567 417 y Fw(in)f(a)g(p)s(oten)m(tial)f Fu(S)6 b Fw(,)33 b(with)g(the)94 533 y(con)m(v)m(en)m(tion)h(that)f Fu(x)849 548 y Fq(b)879 556 y Fg(q)914 548 y Fq(a)951 556 y Fg(p)1019 533 y Fw(=)28 b(0)k(unless)h Fu(b)1533 548 y Fq(q)1572 533 y Fu(a)1623 548 y Fq(p)1695 533 y Fw(is)f(a)g(cyclic)h(2-path.)94 566 y SDict begin H.S end 94 566 a 94 566 a SDict begin 14 H.A end 94 566 a 94 566 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.4.15) cvn H.B /DEST pdfmark end 94 566 a 143 x FA(Prop)s(osition)39 b(4.15.)k Fv(The)36 b(r)-5 b(e)g(duc)g(e)g(d)36 b(p)-5 b(art)37 b Fw(\()p Fu(A)1837 724 y Fp(red)1940 709 y Fu(;)17 b(S)2044 724 y Fp(red)2146 709 y Fw(\))36 b Fv(of)h(a)g(QP)f Fw(\()p Fu(A;)17 b(S)6 b Fw(\))37 b Fv(is)f Fw(2)p Fv(-acyclic)g(if)h(and)94 825 y(only)c(if)f Fu(D)480 789 y Fq(b)510 798 y Fo(1)544 789 y Fq(;:::)o(;b)673 800 y Fg(N)477 849 y Fq(a)514 858 y Fo(1)549 849 y Fq(;:::)n(;a)684 860 y Fg(N)746 825 y Fw(\()p Fu(S)6 b Fw(\))28 b Fr(6)p Fw(=)f(0)32 b Fv(for)h(some)e(c)-5 b(ol)5 b(le)-5 b(ction)32 b(of)g(arr)-5 b(ows)32 b(as)g(ab)-5 b(ove.)43 b(F)-7 b(urthermor)i(e,)32 b(if)g Fu(A)3536 789 y Fn(0)3592 825 y Fv(is)94 941 y(the)25 b(arr)-5 b(ow)25 b(sp)-5 b(an)24 b(of)h(the)g(quiver)f(obtaine)-5 b(d)24 b(fr)-5 b(om)25 b Fu(Q)g Fv(by)g(r)-5 b(emoving)24 b(al)5 b(l)24 b(arr)-5 b(ows)25 b Fu(a)2979 956 y Fp(1)3018 941 y Fu(;)17 b(b)3103 956 y Fp(1)3143 941 y Fu(;)g(:)g(:)g(:)f(;)h(a)3413 956 y Fq(N)3480 941 y Fu(;)g(b)3565 956 y Fq(N)3632 941 y Fv(,)94 1057 y(then)43 b(ther)-5 b(e)44 b(exists)e(a)h(r)-5 b(e)g(gular)44 b(map)e Fu(H)51 b Fw(:)43 b Fu(U)10 b Fw(\()p Fu(D)1893 1021 y Fq(b)1923 1030 y Fo(1)1958 1021 y Fq(;:::)n(;b)2086 1032 y Fg(N)1890 1082 y Fq(a)1927 1091 y Fo(1)1962 1082 y Fq(;:::)o(;a)2098 1093 y Fg(N)2160 1057 y Fw(\))43 b Fr(!)f Fu(K)2473 1021 y Fn(C)t Fp(\()p Fq(A)2594 998 y Ff(0)2617 1021 y Fp(\))2692 1057 y Fv(such)h(that,)j (for)c(any)i(QP)94 1185 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\))53 b Fv(with)g Fu(S)67 b Fr(2)61 b Fu(U)10 b Fw(\()p Fu(D)1088 1149 y Fq(b)1118 1158 y Fo(1)1153 1149 y Fq(;:::)o(;b)1282 1160 y Fg(N)1085 1210 y Fq(a)1122 1219 y Fo(1)1158 1210 y Fq(;:::)n(;a)1293 1221 y Fg(N)1355 1185 y Fw(\))p Fv(,)58 b(the)53 b(r)-5 b(e)g(duc)g(e)g(d)52 b(p)-5 b(art)53 b Fw(\()p Fu(A)2353 1200 y Fp(red)2455 1185 y Fu(;)17 b(S)2559 1200 y Fp(red)2661 1185 y Fw(\))53 b Fv(is)g(right-e)-5 b(quivalent)52 b(to)94 1306 y Fw(\()p Fu(A)205 1270 y Fn(0)229 1306 y Fu(;)17 b(H)8 b Fw(\()p Fu(S)e Fw(\)\))p Fv(.)194 1481 y Fw(The)42 b(pro)s(of)d(of)i(Prop)s(osition)1318 1481 y SDict begin H.S end 1318 1481 a Fw(4.15)1491 1417 y SDict begin H.R end 1491 1417 a 1491 1481 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.15) cvn H.B /ANN pdfmark end 1491 1481 a 41 w Fw(follo)m(ws)e(b)m(y)j(tracing)d(the)i (construction)g(of)f(\()p Fu(A)3315 1496 y Fp(red)3418 1481 y Fu(;)17 b(S)3522 1496 y Fp(red)3623 1481 y Fw(\))94 1597 y(giv)m(en)32 b(in)f(the)i(pro)s(of)d(of)i(Lemma)1340 1597 y SDict begin H.S end 1340 1597 a Fw(4.7)1465 1533 y SDict begin H.R end 1465 1533 a 1465 1597 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.7) cvn H.B /ANN pdfmark end 1465 1597 a Fw(.)43 b(Note)32 b(that)f(w)m(e)i(use)g(the)f (follo)m(wing)d(con)m(v)m(en)m(tion.)44 b(If)32 b Fu(A)g Fw(is)94 1714 y(2-acyclic)j(from)f(the)h(start)h(then)f(the)h(only)f (collection)e Fr(f)p Fu(a)2285 1729 y Fp(1)2325 1714 y Fu(;)17 b(b)2410 1729 y Fp(1)2449 1714 y Fu(;)g(:)g(:)g(:)f(;)h(a) 2719 1729 y Fq(N)2786 1714 y Fu(;)g(b)2871 1729 y Fq(N)2939 1714 y Fr(g)35 b Fw(as)g(ab)s(o)m(v)m(e)h(is)f(the)94 1830 y(empt)m(y)e(set;)g(in)f(this)g(case,)h(the)g(function)f Fu(D)1742 1794 y Fq(b)1772 1803 y Fo(1)1806 1794 y Fq(;:::)o(;b)1935 1805 y Fg(N)1739 1854 y Fq(a)1776 1863 y Fo(1)1811 1854 y Fq(;:::)n(;a)1946 1865 y Fg(N)2040 1830 y Fw(is)g(understo)s(o)s(d)h (to)f(b)s(e)g(equal)g(to)g(1,)h(and)f Fu(H)94 1946 y Fw(is)h(just)g(the)g(iden)m(tit)m(y)f(mapping.)94 2052 y SDict begin H.S end 94 2052 a 94 2052 a SDict begin 14 H.A end 94 2052 a 94 2052 a SDict begin [ /View [/XYZ H.V] /Dest (section.5) cvn H.B /DEST pdfmark end 94 2052 a 844 2148 a Fw(5.)48 b Fx(Mut)-7 b(a)g(tions)38 b(of)h(quivers)e(with)h (potentials)194 2323 y Fw(Let)32 b(\()p Fu(A;)17 b(S)6 b Fw(\))32 b(b)s(e)g(a)g(QP)-8 b(.)33 b(Supp)s(ose)g(that)f(a)g(v)m (ertex)i Fu(k)d Fr(2)d Fu(Q)2288 2338 y Fp(0)2359 2323 y Fw(do)s(es)33 b(not)f(b)s(elong)f(to)h(an)g(orien)m(ted)94 2439 y(2-cycle.)44 b(In)33 b(other)f(w)m(ords,)i Fu(k)i Fw(satis\014es)d(the)g(follo)m(wing)d(condition:)94 2597 y(\(5.1\))929 2597 y SDict begin H.S end 929 2597 a 929 2597 a SDict begin 14 H.A end 929 2597 a 929 2597 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.1) cvn H.B /DEST pdfmark end 929 2597 a Fw(F)-8 b(or)31 b(ev)m(ery)k(v)m(ertex)f Fu(i)p Fw(,)f(either)g Fu(A)2099 2612 y Fq(i;k)2217 2597 y Fw(or)g Fu(A)2410 2612 y Fq(k)r(;i)2529 2597 y Fw(is)f(zero.)94 2756 y(Replacing)j Fu(S)42 b Fw(if)35 b(necessary)k(with)d(a)g (cyclically)e(equiv)-5 b(alen)m(t)36 b(p)s(oten)m(tial,)g(w)m(e)h(can)g (also)e(assume)94 2872 y(that)94 3031 y(\(5.2\))485 3031 y SDict begin H.S end 485 3031 a 485 3031 a SDict begin 14 H.A end 485 3031 a 485 3031 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.2) cvn H.B /DEST pdfmark end 485 3031 a Fw(No)e(cyclic)f(path)g(o)s(ccurring)g(in)g(the)h(expansion)g(of)f Fu(S)39 b Fw(starts)33 b(\(and)f(ends\))i(at)e Fu(k)s Fw(.)94 3203 y(Under)i(these)h(conditions,)e(w)m(e)h(asso)s(ciate)f(to) g(\()p Fu(A;)17 b(S)6 b Fw(\))32 b(a)h(QP)38 b Fm(e)-59 b Fu(\026)2419 3218 y Fq(k)2461 3203 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\))28 b(=)h(\()2913 3178 y Fm(e)2891 3203 y Fu(A;)3021 3178 y Fm(e)3008 3203 y Fu(S)6 b Fw(\))33 b(on)g(the)h(same)94 3334 y(set)g(of)e(v)m(ertices)i Fu(Q)788 3349 y Fp(0)827 3334 y Fw(.)44 b(W)-8 b(e)33 b(de\014ne)h(the)f(homogeneous)f(comp)s(onen)m(ts)2672 3309 y Fm(e)2649 3334 y Fu(A)2722 3349 y Fq(i;j)2835 3334 y Fw(as)h(follo)m(ws:)94 3592 y(\(5.3\))982 3592 y SDict begin H.S end 982 3592 a 982 3592 a SDict begin 14 H.A end 982 3592 a 982 3592 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.3) cvn H.B /DEST pdfmark end 982 3592 a 1004 3567 a Fm(e)982 3592 y Fu(A)1055 3607 y Fq(i;j)1163 3592 y Fw(=)1266 3421 y Fm(\()1346 3510 y Fw(\()p Fu(A)1457 3525 y Fq(j;i)1534 3510 y Fw(\))1572 3474 y Fq(?)2046 3510 y Fw(if)f Fu(i)c Fw(=)f Fu(k)36 b Fw(or)c Fu(j)i Fw(=)27 b Fu(k)s Fw(;)1346 3679 y Fu(A)1419 3694 y Fq(i;j)1522 3679 y Fr(\010)22 b Fu(A)1694 3694 y Fq(i;k)1781 3679 y Fu(A)1854 3694 y Fq(k)r(;j)2046 3679 y Fw(otherwise;)94 3851 y(here)40 b(the)f(pro)s(duct)g Fu(A)931 3866 y Fq(i;k)1017 3851 y Fu(A)1090 3866 y Fq(k)r(;j)1223 3851 y Fw(is)f(understo)s(o)s(d) h(as)g(a)f(subspace)j(of)d Fu(A)2660 3815 y Fp(2)2737 3851 y Fr(\022)h Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fw(.)60 b(Th)m(us,)42 b(the)94 3984 y Fu(R)q Fw(-bimo)s(dule)650 3959 y Fm(e)627 3984 y Fu(A)33 b Fw(is)f(giv)m(en)h(b)m(y)94 4151 y(\(5.4\))1062 4151 y SDict begin H.S end 1062 4151 a 1062 4151 a SDict begin 14 H.A end 1062 4151 a 1062 4151 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.4) cvn H.B /DEST pdfmark end 1062 4151 a 1084 4126 a Fm(e)1062 4151 y Fu(A)28 b Fw(=)p 1266 4096 46 4 v 27 w Fu(e)1311 4166 y Fq(k)1354 4151 y Fu(A)p 1427 4096 V(e)1472 4166 y Fq(k)1537 4151 y Fr(\010)23 b Fu(Ae)1755 4166 y Fq(k)1798 4151 y Fu(A)f Fr(\010)h Fw(\()p Fu(e)2076 4166 y Fq(k)2118 4151 y Fu(A)p Fw(\))2229 4110 y Fq(?)2291 4151 y Fr(\010)g Fw(\()p Fu(Ae)2547 4166 y Fq(k)2590 4151 y Fw(\))2628 4110 y Fq(?)2667 4151 y Fu(;)94 4310 y Fw(where)34 b(w)m(e)g(use)f(the) g(notation)94 4468 y(\(5.5\))1352 4468 y SDict begin H.S end 1352 4468 a 1352 4468 a SDict begin 14 H.A end 1352 4468 a 1352 4468 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.5) cvn H.B /DEST pdfmark end 1352 4468 a 1352 4414 46 4 v Fu(e)1398 4483 y Fq(k)1468 4468 y Fw(=)28 b(1)21 b Fr(\000)i Fu(e)1787 4483 y Fq(k)1858 4468 y Fw(=)2052 4374 y Fm(X)1961 4590 y Fq(i)p Fn(2)p Fq(Q)2088 4599 y Fo(0)2122 4590 y Fn(\000f)p Fq(k)r Fn(g)2303 4468 y Fu(e)2348 4483 y Fq(i)2376 4468 y Fu(:)194 4764 y Fw(W)-8 b(e)36 b(asso)s(ciate)g(to)f Fu(Q)978 4779 y Fp(1)1054 4764 y Fw(the)h(set)h(of)e(arro)m(ws)1829 4739 y Fm(e)1810 4764 y Fu(Q)1887 4779 y Fp(1)1963 4764 y Fw(in)g(the)h(follo)m(wing)d (w)m(a)m(y)-8 b(.)54 b(If)36 b Fu(i)e Fw(=)f Fu(k)39 b Fw(or)c Fu(j)k Fw(=)33 b Fu(k)s Fw(,)94 4880 y(w)m(e)h(set)94 5039 y(\(5.6\))1211 5039 y SDict begin H.S end 1211 5039 a 1211 5039 a SDict begin 14 H.A end 1211 5039 a 1211 5039 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.6) cvn H.B /DEST pdfmark end 1211 5039 a 1230 5014 a Fm(e)1211 5039 y Fu(Q)1288 5054 y Fp(1)1349 5039 y Fr(\\)1460 5014 y Fm(e)1438 5039 y Fu(A)1511 5054 y Fq(i;j)1619 5039 y Fw(=)27 b Fr(f)p Fu(a)1823 4998 y Fq(?)1891 5039 y Fr(j)g Fu(a)h Fr(2)g Fu(Q)2196 5054 y Fp(1)2258 5039 y Fr(\\)22 b Fu(A)2419 5054 y Fq(j;i)2496 5039 y Fr(g)94 5197 y Fw(\(the)33 b(dual)f(basis\).)44 b(And)33 b(for)f Fu(i)g Fw(and)h Fu(j)39 b Fw(di\013eren)m(t)32 b(from)g Fu(k)s Fw(,)g(w)m(e)i(de\014ne)94 5377 y(\(5.7\))549 5377 y SDict begin H.S end 549 5377 a 549 5377 a SDict begin 14 H.A end 549 5377 a 549 5377 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.7) cvn H.B /DEST pdfmark end 549 5377 a 568 5352 a Fm(e)549 5377 y Fu(Q)626 5392 y Fp(1)688 5377 y Fr(\\)799 5352 y Fm(e)776 5377 y Fu(A)849 5392 y Fq(i;j)957 5377 y Fw(=)28 b(\()p Fu(Q)1176 5392 y Fp(1)1237 5377 y Fr(\\)23 b Fu(A)1399 5392 y Fq(i;j)1479 5377 y Fw(\))1566 5283 y Fm(G)1710 5377 y Fr(f)p Fu(ba)28 b Fr(j)f Fu(b)h Fr(2)g Fu(Q)2175 5392 y Fp(1)2237 5377 y Fr(\\)23 b Fu(A)2399 5392 y Fq(i;k)2485 5377 y Fu(;)50 b(a)28 b Fr(2)g Fu(Q)2812 5392 y Fp(1)2874 5377 y Fr(\\)22 b Fu(A)3035 5392 y Fq(k)r(;j)3130 5377 y Fr(g)p Fw(;)94 5580 y(to)28 b(a)m(v)m(oid)g(confusion,)g(w)m(e)h(denote)g(b)m(y)f([)p Fu(ba)p Fw(])h Fr(2)1783 5555 y Fm(e)1764 5580 y Fu(Q)1841 5595 y Fp(1)1893 5580 y Fr(\\)12 b Fu(A)2044 5595 y Fq(i;k)2131 5580 y Fu(A)2204 5595 y Fq(k)r(;j)2327 5580 y Fw(the)28 b(arro)m(w)g(in)2887 5555 y Fm(e)2868 5580 y Fu(Q)2945 5595 y Fp(1)3012 5580 y Fw(asso)s(ciated)g(with)94 5696 y(the)33 b(pro)s(duct)g Fu(ba)p Fw(.)p eop end end %%Page: 21 21 TeXDict begin HPSdict begin 21 20 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.21) cvn H.B /DEST pdfmark end 94 218 a 1179 w Fs(QUIVERS)31 b(WITH)i(POTENTIALS)g(I)1102 b(21)194 431 y Fw(W)-8 b(e)33 b(no)m(w)g(asso)s(ciate)g(to)f Fu(S)38 b Fw(the)33 b(p)s(oten)m(tial)i Fm(e)-59 b Fu(\026)1832 446 y Fq(k)1874 431 y Fw(\()p Fu(S)6 b Fw(\))28 b(=)2161 406 y Fm(e)2147 431 y Fu(S)34 b Fr(2)28 b Fu(R)q Fr(hh)2510 406 y Fm(e)2488 431 y Fu(A)o Fr(ii)k Fw(giv)m(en)h(b)m(y)94 602 y(\(5.8\))1584 602 y SDict begin H.S end 1584 602 a 1584 602 a SDict begin 14 H.A end 1584 602 a 1584 602 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.8) cvn H.B /DEST pdfmark end 1584 602 a 1597 577 a Fm(e)1584 602 y Fu(S)g Fw(=)28 b([)p Fu(S)6 b Fw(])22 b(+)g(\001)2102 617 y Fq(k)2145 602 y Fu(;)94 759 y Fw(where)94 916 y(\(5.9\))1041 916 y SDict begin H.S end 1041 916 a 1041 916 a SDict begin 14 H.A end 1041 916 a 1041 916 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.9) cvn H.B /DEST pdfmark end 1041 916 a Fw(\001)1122 931 y Fq(k)1193 916 y Fw(=)27 b(\001)1377 931 y Fq(k)1420 916 y Fw(\()p Fu(A)p Fw(\))h(=)1963 821 y Fm(X)1701 1037 y Fq(a;b)p Fn(2)p Fq(Q)1891 1046 y Fo(1)1925 1037 y Fp(:)k Fq(h)p Fp(\()p Fq(a)p Fp(\)=)p Fq(t)p Fp(\()p Fq(b)p Fp(\)=)p Fq(k)2369 916 y Fw([)p Fu(ba)p Fw(])p Fu(a)2566 875 y Fq(?)2607 916 y Fu(b)2648 875 y Fq(?)2688 916 y Fu(;)94 1187 y Fw(and)48 b([)p Fu(S)6 b Fw(])48 b(is)f(obtained)h(b)m(y)g(substituting)g([)p Fu(a)1784 1202 y Fq(p)1824 1187 y Fu(a)1875 1202 y Fq(p)p Fp(+1)2005 1187 y Fw(])g(for)f(eac)m(h)i(factor)e Fu(a)2824 1202 y Fq(p)2864 1187 y Fu(a)2915 1202 y Fq(p)p Fp(+1)3093 1187 y Fw(with)g Fu(t)p Fw(\()p Fu(a)3454 1202 y Fq(p)3494 1187 y Fw(\))54 b(=)94 1304 y Fu(h)p Fw(\()p Fu(a)239 1319 y Fq(p)p Fp(+1)370 1304 y Fw(\))30 b(=)h Fu(k)37 b Fw(of)d(an)m(y)h(cyclic)f(path)g Fu(a)1482 1319 y Fp(1)1538 1304 y Fr(\001)17 b(\001)g(\001)e Fu(a)1722 1319 y Fq(d)1797 1304 y Fw(o)s(ccurring)34 b(in)f(the)i(expansion)g(of)f Fu(S)40 b Fw(\(recall)33 b(that)94 1420 y(none)f(of)f(these)h(cyclic)f (paths)g(starts)h(at)f Fu(k)s Fw(\).)43 b(It)31 b(is)f(easy)j(to)d(see) j(that)d(b)s(oth)h([)p Fu(S)6 b Fw(])31 b(and)h(\001)3313 1435 y Fq(k)3387 1420 y Fw(do)f(not)94 1536 y(dep)s(end)j(on)f(the)g(c) m(hoice)g(of)f(a)g(basis)h Fu(Q)1536 1551 y Fp(1)1608 1536 y Fw(of)f Fu(A)p Fw(.)194 1652 y(The)i(follo)m(wing)c(prop)s (osition)g(is)i(immediate)e(from)i(the)h(de\014nitions.)94 1672 y SDict begin H.S end 94 1672 a 94 1672 a SDict begin 14 H.A end 94 1672 a 94 1672 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.5.1) cvn H.B /DEST pdfmark end 94 1672 a 147 x FA(Prop)s(osition)k(5.1.)42 b Fv(Supp)-5 b(ose)35 b(a)h(QP)f Fw(\()p Fu(A;)17 b(S)6 b Fw(\))35 b Fv(satis\014es)g Fw(\()2250 1819 y SDict begin H.S end 2250 1819 a Fw(5.1)2375 1755 y SDict begin H.R end 2375 1755 a 2375 1819 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.1) cvn H.B /ANN pdfmark end 2375 1819 a Fw(\))g Fv(and)g Fw(\()2676 1819 y SDict begin H.S end 2676 1819 a Fw(5.2)2801 1755 y SDict begin H.R end 2801 1755 a 2801 1819 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.2) cvn H.B /ANN pdfmark end 2801 1819 a Fw(\))p Fv(,)g(and)g(a)h(QP)f Fw(\()p Fu(A)3467 1783 y Fn(0)3491 1819 y Fu(;)17 b(S)3601 1783 y Fn(0)3623 1819 y Fw(\))94 1935 y Fv(is)35 b(such)g(that)g Fu(e)665 1950 y Fq(k)708 1935 y Fu(A)781 1899 y Fn(0)832 1935 y Fw(=)28 b Fu(A)1009 1899 y Fn(0)1032 1935 y Fu(e)1077 1950 y Fq(k)1148 1935 y Fw(=)f Fr(f)p Fw(0)p Fr(g)p Fv(.)44 b(Then)34 b(we)h(have)94 2092 y Fw(\(5.10\))1012 2092 y SDict begin H.S end 1012 2092 a 1012 2092 a SDict begin 14 H.A end 1012 2092 a 1012 2092 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.10) cvn H.B /DEST pdfmark end 1012 2092 a 4 w Fm(e)-59 b Fu(\026)1071 2107 y Fq(k)1113 2092 y Fw(\()p Fu(A)22 b Fr(\010)h Fu(A)1419 2051 y Fn(0)1442 2092 y Fu(;)17 b(S)28 b Fw(+)22 b Fu(S)1738 2051 y Fn(0)1761 2092 y Fw(\))28 b(=)k Fm(e)-60 b Fu(\026)1989 2107 y Fq(k)2032 2092 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\))21 b Fr(\010)i Fw(\()p Fu(A)2523 2051 y Fn(0)2546 2092 y Fu(;)17 b(S)2656 2051 y Fn(0)2679 2092 y Fw(\))p Fu(:)94 2167 y SDict begin H.S end 94 2167 a 94 2167 a SDict begin 14 H.A end 94 2167 a 94 2167 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.5.2) cvn H.B /DEST pdfmark end 94 2167 a 106 x FA(Theorem)48 b(5.2.)f Fv(The)c(right-e)-5 b(quivalenc)g(e)42 b(class)g(of)h(the)h(QP)f Fw(\()2536 2247 y Fm(e)2514 2273 y Fu(A;)2644 2247 y Fm(e)2631 2273 y Fu(S)6 b Fw(\))43 b(=)48 b Fm(e)-60 b Fu(\026)2956 2288 y Fq(k)2999 2273 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\))42 b Fv(is)i(deter-)94 2389 y(mine)-5 b(d)34 b(by)h(the)g(right-e)-5 b(quivalenc)g(e)34 b(class)g(of)h Fw(\()p Fu(A;)17 b(S)6 b Fw(\))p Fv(.)94 2570 y(Pr)-5 b(o)g(of.)42 b Fw(Let)590 2544 y Fm(b)567 2570 y Fu(A)33 b Fw(b)s(e)g(the)g(\014nite-dimensional)c Fu(R)q Fw(-bimo)s(dule)h(giv)m(en)j(b)m(y)94 2735 y(\(5.11\))1328 2735 y SDict begin H.S end 1328 2735 a 1328 2735 a SDict begin 14 H.A end 1328 2735 a 1328 2735 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.11) cvn H.B /DEST pdfmark end 1328 2735 a 1350 2710 a Fm(b)1328 2735 y Fu(A)27 b Fw(=)h Fu(A)22 b Fr(\010)h Fw(\()p Fu(e)1810 2750 y Fq(k)1853 2735 y Fu(A)p Fw(\))1964 2694 y Fq(?)2025 2735 y Fr(\010)g Fw(\()p Fu(Ae)2281 2750 y Fq(k)2324 2735 y Fw(\))2362 2694 y Fq(?)2401 2735 y Fu(:)94 2906 y Fw(The)28 b(natural)d(em)m(b)s (edding)g Fu(A)j Fr(!)1361 2881 y Fm(b)1338 2906 y Fu(A)f Fw(iden)m(ti\014es)f Fu(R)q Fr(hh)p Fu(A)p Fr(ii)f Fw(with)h(a)g (closed)g(subalgebra)g(in)f Fu(R)q Fr(hh)3506 2881 y Fm(b)3484 2906 y Fu(A)p Fr(ii)p Fw(.)94 3036 y(W)-8 b(e)45 b(also)f(ha)m(v)m(e)h(a)f(natural)f(em)m(b)s(edding)1692 3011 y Fm(e)1669 3036 y Fu(A)48 b Fr(!)f Fu(R)q Fr(hh)2112 3011 y Fm(b)2090 3036 y Fu(A)p Fr(ii)d Fw(\(sending)g(eac)m(h)h(arro)m (w)g([)p Fu(ba)p Fw(])g(to)f(the)94 3167 y(pro)s(duct)d Fu(ba)p Fw(\).)69 b(This)41 b(allo)m(ws)f(us)h(to)f(iden)m(tify)h Fu(R)q Fr(hh)2023 3142 y Fm(e)2001 3167 y Fu(A)o Fr(ii)f Fw(with)h(another)g(closed)g(subalgebra)f(in)94 3297 y Fu(R)q Fr(hh)269 3272 y Fm(b)247 3297 y Fu(A)p Fr(ii)p Fw(,)c(namely)-8 b(,)36 b(with)f(the)h(closure)g(of)f(the)h(linear)e (span)i(of)f(the)h(paths)e Fm(b)-53 b Fu(a)2943 3312 y Fp(1)2999 3297 y Fr(\001)17 b(\001)g(\001)c Fm(b)-53 b Fu(a)3183 3312 y Fq(d)3260 3297 y Fw(suc)m(h)37 b(that)92 3427 y Fm(b)-53 b Fu(a)145 3442 y Fp(1)224 3427 y Fu(=)-60 b Fr(2)28 b Fu(e)352 3442 y Fq(k)395 3427 y Fu(A)h Fw(and)f Fm(b)-53 b Fu(a)735 3442 y Fq(d)815 3427 y Fu(=)-60 b Fr(2)28 b Fu(Ae)1016 3442 y Fq(k)1059 3427 y Fw(.)42 b(Under)30 b(this)g(iden)m(ti\014cation,)f(the)h(p)s(oten)m(tial)2808 3402 y Fm(e)2794 3427 y Fu(S)36 b Fw(giv)m(en)30 b(b)m(y)g(\()3312 3427 y SDict begin H.S end 3312 3427 a Fw(5.8)3437 3363 y SDict begin H.R end 3437 3363 a 3437 3427 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.8) cvn H.B /ANN pdfmark end 3437 3427 a Fw(\))f(and)94 3558 y(view)m(ed)34 b(as)f(an)f(elemen)m(t)h(of)f Fu(R)q Fr(hh)1316 3533 y Fm(b)1294 3558 y Fu(A)p Fr(ii)g Fw(is)g(cyclically)e(equiv)-5 b(alen)m(t)33 b(to)f(the)h(p)s(oten)m(tial)1205 3736 y Fu(S)28 b Fw(+)22 b(\()1526 3642 y Fm(X)1429 3854 y Fq(b)p Fn(2)p Fq(Q)1562 3863 y Fo(1)1596 3854 y Fn(\\)p Fq(Ae)1729 3866 y Fg(k)1784 3736 y Fu(b)1825 3695 y Fq(?)1865 3736 y Fu(b)p Fw(\)\()2083 3642 y Fm(X)1982 3853 y Fq(a)p Fn(2)p Fq(Q)2122 3862 y Fo(1)2157 3853 y Fn(\\)p Fq(e)2237 3865 y Fg(k)2274 3853 y Fq(A)2344 3736 y Fu(aa)2446 3695 y Fq(?)2486 3736 y Fw(\))p Fu(:)194 3995 y Fw(T)-8 b(aking)34 b(this)f(in)m(to)g(accoun)m(t,)j(w)m(e)f(see)g(that)f(Theorem)2231 3995 y SDict begin H.S end 2231 3995 a Fw(5.2)2355 3931 y SDict begin H.R end 2355 3931 a 2355 3995 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.5.2) cvn H.B /ANN pdfmark end 2355 3995 a 34 w Fw(b)s(ecomes)h(a)e(consequence)k(of)d(the)94 4111 y(follo)m(wing)c(lemma.)94 4130 y SDict begin H.S end 94 4130 a 94 4130 a SDict begin 14 H.A end 94 4130 a 94 4130 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.5.3) cvn H.B /DEST pdfmark end 94 4130 a 148 x FA(Lemma)g(5.3.)37 b Fv(Every)29 b(automorphism)f Fu(')h Fv(of)g Fu(R)q Fr(hh)p Fu(A)p Fr(ii)f Fv(c)-5 b(an)29 b(b)-5 b(e)28 b(extende)-5 b(d)29 b(to)g(an)f(automorphism)106 4408 y Fm(b)-67 b Fu(')35 b Fv(of)g Fu(R)q Fr(hh)483 4383 y Fm(b)461 4408 y Fu(A)p Fr(ii)f Fv(satisfying)94 4579 y Fw(\(5.12\))1426 4579 y SDict begin H.S end 1426 4579 a 1426 4579 a SDict begin 14 H.A end 1426 4579 a 1426 4579 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.12) cvn H.B /DEST pdfmark end 1426 4579 a 12 w Fm(b)-67 b Fu(')p Fw(\()p Fu(R)q Fr(hh)1703 4554 y Fm(e)1681 4579 y Fu(A)o Fr(ii)p Fw(\))27 b(=)h Fu(R)q Fr(hh)2175 4554 y Fm(e)2153 4579 y Fu(A)p Fr(ii)p Fu(;)94 4736 y Fv(and)94 4893 y Fw(\(5.13\))665 4893 y SDict begin H.S end 665 4893 a 665 4893 a SDict begin 14 H.A end 665 4893 a 665 4893 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.13) cvn H.B /DEST pdfmark end 665 4893 a 13 w Fm(b)-68 b Fu(')p Fw(\()868 4798 y Fm(X)767 5010 y Fq(a)p Fn(2)p Fq(Q)907 5019 y Fo(1)942 5010 y Fn(\\)p Fq(e)1022 5022 y Fg(k)1060 5010 y Fq(A)1129 4893 y Fu(aa)1231 4852 y Fq(?)1271 4893 y Fw(\))28 b(=)1541 4798 y Fm(X)1440 5010 y Fq(a)p Fn(2)p Fq(Q)1580 5019 y Fo(1)1615 5010 y Fn(\\)p Fq(e)1695 5022 y Fg(k)1733 5010 y Fq(A)1802 4893 y Fu(aa)1904 4852 y Fq(?)1944 4893 y Fu(;)129 b Fm(b)-68 b Fu(')p Fw(\()2286 4798 y Fm(X)2189 5010 y Fq(b)p Fn(2)p Fq(Q)2322 5019 y Fo(1)2357 5010 y Fn(\\)p Fq(Ae)2490 5022 y Fg(k)2544 4893 y Fu(b)2585 4852 y Fq(?)2625 4893 y Fu(b)p Fw(\))28 b(=)2933 4798 y Fm(X)2836 5010 y Fq(b)p Fn(2)p Fq(Q)2969 5019 y Fo(1)3003 5010 y Fn(\\)p Fq(Ae)3136 5022 y Fg(k)3190 4893 y Fu(b)3231 4852 y Fq(?)3271 4893 y Fu(b:)194 5189 y Fw(In)j(order)f(to)g(extend)i Fu(')f Fw(to)f(an)g(automorphism)41 b Fm(b)-68 b Fu(')30 b Fw(of)g Fu(R)q Fr(hh)2364 5164 y Fm(b)2342 5189 y Fu(A)p Fr(ii)p Fw(,)g(w)m(e)i(need)f(only)f(to)g (de\014ne)i(the)94 5305 y(elemen)m(ts)46 b Fm(b)-68 b Fu(')p Fw(\()p Fu(a)646 5269 y Fq(?)686 5305 y Fw(\))32 b(and)45 b Fm(b)-67 b Fu(')p Fw(\()p Fu(b)1089 5269 y Fq(?)1129 5305 y Fw(\))32 b(for)g(all)f(arro)m(ws)i Fu(a)28 b Fr(2)g Fu(Q)2046 5320 y Fp(1)2108 5305 y Fr(\\)22 b Fu(e)2241 5320 y Fq(k)2284 5305 y Fu(A)33 b Fw(and)f Fu(b)d Fr(2)f Fu(Q)2820 5320 y Fp(1)2881 5305 y Fr(\\)23 b Fu(Ae)3088 5320 y Fq(k)3131 5305 y Fw(.)194 5421 y(W)-8 b(e)31 b(\014rst)g(deal)f(with)43 b Fm(b)-68 b Fu(')p Fw(\()p Fu(a)1136 5385 y Fq(?)1176 5421 y Fw(\).)43 b(Let)31 b Fu(Q)1534 5436 y Fp(1)1591 5421 y Fr(\\)19 b Fu(e)1721 5436 y Fq(k)1764 5421 y Fu(A)28 b Fw(=)f Fr(f)p Fu(a)2069 5436 y Fp(1)2109 5421 y Fu(;)17 b(:)g(:)g(:)e(;)i(a)2378 5436 y Fq(s)2415 5421 y Fr(g)p Fw(.)43 b(In)31 b(view)g(of)f(Prop)s (osition)3510 5421 y SDict begin H.S end 3510 5421 a Fw(2.5)3634 5357 y SDict begin H.R end 3634 5357 a 3634 5421 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.2.5) cvn H.B /ANN pdfmark end 3634 5421 a Fw(,)94 5538 y(the)j(action)f(of)g Fu(')h Fw(on)f(these)i(arro)m(ws)f(is)f(giv)m(en)h(b)m(y)94 5695 y(\(5.14\))623 5695 y SDict begin H.S end 623 5695 a 623 5695 a SDict begin 14 H.A end 623 5695 a 623 5695 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.14) cvn H.B /DEST pdfmark end 623 5695 a 639 5614 a Fm(\000)685 5693 y Fu(')p Fw(\()p Fu(a)838 5708 y Fp(1)878 5693 y Fw(\))82 b Fu(')p Fw(\()p Fu(a)1151 5708 y Fp(2)1191 5693 y Fw(\))h Fr(\001)17 b(\001)g(\001)98 b Fu(')p Fw(\()p Fu(a)1681 5708 y Fq(s)1718 5693 y Fw(\))1756 5614 y Fm(\001)1829 5695 y Fw(=)1932 5614 y Fm(\000)1978 5693 y Fu(a)2029 5708 y Fp(1)2152 5693 y Fu(a)2203 5708 y Fp(2)2325 5693 y Fr(\001)17 b(\001)g(\001)98 b Fu(a)2592 5708 y Fq(s)2629 5614 y Fm(\001)2692 5695 y Fw(\()p Fu(C)2800 5710 y Fp(0)2861 5695 y Fw(+)22 b Fu(C)3029 5710 y Fp(1)3068 5695 y Fw(\))p Fu(;)p eop end end %%Page: 22 22 TeXDict begin HPSdict begin 22 21 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.22) cvn H.B /DEST pdfmark end 94 218 a Fs(22)492 b(HARM)25 b(DERKSEN,)f(JERZY)i(WEYMAN,)f(AND)f(ANDREI)g(ZELEVINSKY)94 417 y Fw(where:)318 587 y Fr(\017)41 b Fu(C)479 602 y Fp(0)556 587 y Fw(is)c(an)h(in)m(v)m(ertible)f Fu(s)26 b Fr(\002)g Fu(s)37 b Fw(matrix)g(with)g(en)m(tries)h(in)f Fu(K)45 b Fw(suc)m(h)40 b(that)d(its)g(\()p Fu(p;)17 b(q)t Fw(\)-en)m(try)409 703 y(is)32 b(0)g(unless)i Fu(t)p Fw(\()p Fu(a)1001 718 y Fq(p)1041 703 y Fw(\))27 b(=)h Fu(t)p Fw(\()p Fu(a)1334 718 y Fq(q)1372 703 y Fw(\);)318 820 y Fr(\017)41 b Fu(C)479 835 y Fp(1)551 820 y Fw(is)32 b(a)g Fu(s)22 b Fr(\002)h Fu(s)32 b Fw(matrix)f(whose)j(\()p Fu(p;)17 b(q)t Fw(\)-en)m(try)33 b(b)s(elongs)f(to)g Fl(m)p Fw(\()p Fu(A)p Fw(\))2778 835 y Fq(t)p Fp(\()p Fq(a)2867 843 y Fg(p)2905 835 y Fp(\))p Fq(;t)p Fp(\()p Fq(a)3041 843 y Fg(q)3077 835 y Fp(\))3108 820 y Fw(.)94 990 y(Note)g(that)g Fu(C)610 1005 y Fp(0)669 990 y Fw(+)20 b Fu(C)835 1005 y Fp(1)906 990 y Fw(is)31 b(in)m(v)m(ertible,)g(and)h (its)f(in)m(v)m(erse)i(is)e(of)g(the)h(same)f(form:)42 b(indeed,)32 b(w)m(e)h(ha)m(v)m(e)475 1288 y(\()p Fu(C)583 1303 y Fp(0)644 1288 y Fw(+)22 b Fu(C)812 1303 y Fp(1)852 1288 y Fw(\))890 1247 y Fn(\000)p Fp(1)1012 1288 y Fw(=)27 b(\()p Fu(I)j Fw(+)22 b Fu(C)1401 1247 y Fn(\000)p Fp(1)1394 1313 y(0)1495 1288 y Fu(C)1565 1303 y Fp(1)1605 1288 y Fw(\))1643 1247 y Fn(\000)p Fp(1)1737 1288 y Fu(C)1814 1247 y Fn(\000)p Fp(1)1807 1313 y(0)1936 1288 y Fw(=)27 b(\()p Fu(I)j Fw(+)2285 1164 y Fn(1)2248 1194 y Fm(X)2254 1403 y Fq(n)p Fp(=1)2392 1288 y Fw(\()p Fr(\000)p Fw(1\))2594 1247 y Fq(n)2641 1288 y Fw(\()p Fu(C)2756 1247 y Fn(\000)p Fp(1)2749 1313 y(0)2850 1288 y Fu(C)2920 1303 y Fp(1)2960 1288 y Fw(\))2998 1247 y Fq(n)3045 1288 y Fw(\))p Fu(C)3160 1247 y Fn(\000)p Fp(1)3153 1313 y(0)3254 1288 y Fu(:)94 1604 y Fw(No)m(w)k(w)m(e)f(de\014ne)h(the)f(elemen)m(ts)45 b Fm(b)-67 b Fu(')p Fw(\()p Fu(a)1462 1567 y Fq(?)1462 1628 y(p)1502 1604 y Fw(\))32 b(b)m(y)i(setting)1193 1777 y Fm(0)1193 1952 y(B)1193 2012 y(B)1193 2076 y(@)1292 1839 y(b)-67 b Fu(')p Fw(\()p Fu(a)1433 1802 y Fq(?)1433 1863 y Fp(1)1473 1839 y Fw(\))1292 1955 y Fm(b)g Fu(')p Fw(\()p Fu(a)1433 1919 y Fq(?)1433 1979 y Fp(2)1473 1955 y Fw(\))1382 2049 y(.)1382 2082 y(.)1382 2116 y(.)1292 2232 y Fm(b)g Fu(')p Fw(\()p Fu(a)1433 2196 y Fq(?)1433 2256 y(s)1473 2232 y Fw(\))1511 1777 y Fm(1)1511 1952 y(C)1511 2012 y(C)1511 2076 y(A)1625 2037 y Fw(=)28 b(\()p Fu(C)1837 2052 y Fp(0)1898 2037 y Fw(+)22 b Fu(C)2066 2052 y Fp(1)2105 2037 y Fw(\))2143 1996 y Fn(\000)p Fp(1)2254 1777 y Fm(0)2254 1952 y(B)2254 2012 y(B)2254 2076 y(@)2341 1839 y Fu(a)2392 1802 y Fq(?)2392 1863 y Fp(1)2341 1955 y Fu(a)2392 1919 y Fq(?)2392 1979 y Fp(2)2373 2049 y Fw(.)2373 2082 y(.)2373 2116 y(.)2341 2232 y Fu(a)2392 2196 y Fq(?)2392 2256 y(s)2432 1777 y Fm(1)2432 1952 y(C)2432 2012 y(C)2432 2076 y(A)2536 2037 y Fu(:)94 2452 y Fw(It)33 b(follo)m(ws)e(that)792 2848 y Fm(b)-67 b Fu(')p Fw(\()882 2753 y Fm(X)936 2962 y Fq(p)1042 2848 y Fu(a)1093 2863 y Fq(p)1133 2848 y Fu(a)1184 2807 y Fq(?)1184 2872 y(p)1224 2848 y Fw(\))27 b(=)1393 2767 y Fm(\000)1451 2846 y(b)-67 b Fu(')p Fw(\()p Fu(a)1592 2861 y Fp(1)1631 2846 y Fw(\))95 b Fm(b)-67 b Fu(')p Fw(\()p Fu(a)1905 2861 y Fp(2)1945 2846 y Fw(\))83 b Fr(\001)17 b(\001)g(\001)110 b Fm(b)-68 b Fu(')p Fw(\()p Fu(a)2434 2861 y Fq(s)2471 2846 y Fw(\))2509 2767 y Fm(\001)2571 2588 y(0)2571 2763 y(B)2571 2823 y(B)2571 2887 y(@)2671 2649 y(b)h Fu(')o Fw(\()p Fu(a)2811 2613 y Fq(?)2811 2674 y Fp(1)2851 2649 y Fw(\))2671 2766 y Fm(b)g Fu(')o Fw(\()p Fu(a)2811 2730 y Fq(?)2811 2790 y Fp(2)2851 2766 y Fw(\))2760 2860 y(.)2760 2893 y(.)2760 2926 y(.)2671 3043 y Fm(b)g Fu(')o Fw(\()p Fu(a)2811 3007 y Fq(?)2811 3067 y(s)2851 3043 y Fw(\))2889 2588 y Fm(1)2889 2763 y(C)2889 2823 y(C)2889 2887 y(A)1289 3395 y Fw(=)1393 3314 y Fm(\000)1439 3393 y Fu(a)1490 3408 y Fp(1)1612 3393 y Fu(a)1663 3408 y Fp(2)1786 3393 y Fr(\001)17 b(\001)g(\001)98 b Fu(a)2053 3408 y Fq(s)2090 3314 y Fm(\001)2152 3135 y(0)2152 3310 y(B)2152 3370 y(B)2152 3433 y(@)2239 3196 y Fu(a)2290 3160 y Fq(?)2290 3221 y Fp(1)2239 3313 y Fu(a)2290 3276 y Fq(?)2290 3337 y Fp(2)2271 3407 y Fw(.)2271 3440 y(.)2271 3473 y(.)2239 3590 y Fu(a)2290 3553 y Fq(?)2290 3614 y(s)2330 3135 y Fm(1)2330 3310 y(C)2330 3370 y(C)2330 3433 y(A)2445 3395 y Fw(=)2548 3300 y Fm(X)2602 3509 y Fq(p)2709 3395 y Fu(a)2760 3410 y Fq(p)2800 3395 y Fu(a)2851 3353 y Fq(?)2851 3419 y(p)2891 3395 y Fu(:)194 3815 y Fw(F)-8 b(or)44 b Fu(b)k Fr(2)g Fu(Q)661 3830 y Fp(1)731 3815 y Fr(\\)30 b Fu(Ae)945 3830 y Fq(k)988 3815 y Fw(,)47 b(w)m(e)f(de\014ne)58 b Fm(b)-68 b Fu(')p Fw(\()p Fu(b)1654 3779 y Fq(?)1694 3815 y Fw(\))44 b(in)g(a)g(similar)d (w)m(a)m(y)-8 b(.)80 b(Namely)-8 b(,)46 b(let)e Fu(Q)3211 3830 y Fp(1)3280 3815 y Fr(\\)31 b Fu(Ae)3495 3830 y Fq(k)3586 3815 y Fw(=)94 3931 y Fr(f)p Fu(b)185 3946 y Fp(1)225 3931 y Fu(;)17 b(:)g(:)g(:)f(;)h(b)485 3946 y Fq(t)515 3931 y Fr(g)p Fw(.)43 b(As)33 b(ab)s(o)m(v)m(e,)h(the)f (action)e(of)h Fu(')h Fw(on)f(these)i(arro)m(ws)f(is)f(giv)m(en)h(b)m (y)94 4352 y(\(5.15\))1230 4352 y SDict begin H.S end 1230 4352 a 1230 4352 a SDict begin 14 H.A end 1230 4352 a 1230 4352 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.15) cvn H.B /DEST pdfmark end 1230 4352 a 1247 4092 a Fm(0)1247 4267 y(B)1247 4327 y(B)1247 4391 y(@)1334 4154 y Fu(')p Fw(\()p Fu(b)1477 4169 y Fp(1)1517 4154 y Fw(\))1334 4270 y Fu(')p Fw(\()p Fu(b)1477 4285 y Fp(2)1517 4270 y Fw(\))1431 4364 y(.)1431 4398 y(.)1431 4431 y(.)1339 4547 y Fu(')p Fw(\()p Fu(b)1482 4562 y Fq(t)1512 4547 y Fw(\))1555 4092 y Fm(1)1555 4267 y(C)1555 4327 y(C)1555 4391 y(A)1670 4352 y Fw(=)27 b(\()p Fu(D)1892 4367 y Fp(0)1954 4352 y Fw(+)22 b Fu(D)2133 4367 y Fp(1)2172 4352 y Fw(\))2227 4092 y Fm(0)2227 4267 y(B)2227 4327 y(B)2227 4391 y(@)2314 4154 y Fu(b)2355 4169 y Fp(1)2314 4270 y Fu(b)2355 4285 y Fp(2)2341 4364 y Fw(.)2341 4398 y(.)2341 4431 y(.)2319 4547 y Fu(b)2360 4562 y Fq(t)2395 4092 y Fm(1)2395 4267 y(C)2395 4327 y(C)2395 4391 y(A)2499 4352 y Fu(;)94 4767 y Fw(where:)318 4937 y Fr(\017)41 b Fu(D)490 4952 y Fp(0)568 4937 y Fw(is)d(an)g(in)m(v)m(ertible)g Fu(t)26 b Fr(\002)h Fu(t)39 b Fw(matrix)e(with)h(en)m(tries)h(in)f Fu(K)45 b Fw(suc)m(h)40 b(that)e(its)g(\()p Fu(p;)17 b(q)t Fw(\)-en)m(try)409 5054 y(is)32 b(0)g(unless)i Fu(h)p Fw(\()p Fu(b)1012 5069 y Fq(p)1052 5054 y Fw(\))28 b(=)f Fu(h)p Fw(\()p Fu(b)1356 5069 y Fq(q)1395 5054 y Fw(\);)318 5170 y Fr(\017)41 b Fu(D)490 5185 y Fp(1)562 5170 y Fw(is)32 b(a)g Fu(t)23 b Fr(\002)f Fu(t)33 b Fw(matrix)e(whose)j (\()p Fu(p;)17 b(q)t Fw(\)-en)m(try)33 b(b)s(elongs)f(to)g Fl(m)p Fw(\()p Fu(A)p Fw(\))2768 5185 y Fq(h)p Fp(\()p Fq(b)2866 5193 y Fg(p)2903 5185 y Fp(\))p Fq(;h)p Fp(\()p Fq(b)3048 5193 y Fg(q)3082 5185 y Fp(\))3114 5170 y Fw(.)94 5340 y(As)e(ab)s(o)m(v)m(e,)g(w)m(e)g(see)g(that)f Fu(D)1118 5355 y Fp(0)1172 5340 y Fw(+)14 b Fu(D)1343 5355 y Fp(1)1411 5340 y Fw(is)29 b(in)m(v)m(ertible,)g(and)g(its)f(in)m(v)m(erse)i(is)e (of)h(the)g(same)g(form.)41 b(No)m(w)94 5456 y(w)m(e)34 b(de\014ne)g(the)f(elemen)m(ts)45 b Fm(b)-67 b Fu(')p Fw(\()p Fu(b)1230 5420 y Fq(?)1230 5481 y(q)1270 5456 y Fw(\))32 b(b)m(y)h(setting)599 5614 y Fm(\000)657 5693 y(b)-67 b Fu(')p Fw(\()p Fu(b)788 5657 y Fq(?)788 5717 y Fp(1)828 5693 y Fw(\))95 b Fm(b)-67 b Fu(')p Fw(\()p Fu(b)1092 5657 y Fq(?)1092 5717 y Fp(2)1131 5693 y Fw(\))83 b Fr(\001)17 b(\001)g(\001)111 b Fm(b)-68 b Fu(')p Fw(\()p Fu(b)1611 5657 y Fq(?)1611 5717 y(t)1651 5693 y Fw(\))1689 5614 y Fm(\001)1762 5695 y Fw(=)1866 5614 y Fm(\000)1911 5693 y Fu(b)1952 5657 y Fq(?)1952 5717 y Fp(1)2075 5693 y Fu(b)2116 5657 y Fq(?)2116 5717 y Fp(2)2239 5693 y Fr(\001)17 b(\001)g(\001)98 b Fu(b)2496 5657 y Fq(?)2496 5717 y(t)2536 5614 y Fm(\001)2598 5695 y Fw(\()p Fu(D)2717 5710 y Fp(0)2779 5695 y Fw(+)22 b Fu(D)2958 5710 y Fp(1)2997 5695 y Fw(\))3035 5653 y Fn(\000)p Fp(1)3130 5695 y Fu(:)p eop end end %%Page: 23 23 TeXDict begin HPSdict begin 23 22 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.23) cvn H.B /DEST pdfmark end 94 218 a 1179 w Fs(QUIVERS)31 b(WITH)i(POTENTIALS)g(I)1102 b(23)94 417 y Fw(It)33 b(follo)m(ws)e(that)801 783 y Fm(b)-67 b Fu(')p Fw(\()891 689 y Fm(X)946 898 y Fq(q)1051 783 y Fu(b)1092 742 y Fq(?)1092 808 y(q)1132 783 y Fu(b)1173 798 y Fq(q)1212 783 y Fw(\))27 b(=)1381 703 y Fm(\000)1439 781 y(b)-68 b Fu(')p Fw(\()p Fu(b)1569 796 y Fp(1)1609 781 y Fw(\))1647 746 y Fq(?)1782 781 y Fm(b)h Fu(')o Fw(\()p Fu(b)1912 746 y Fq(?)1912 806 y Fp(2)1952 781 y Fw(\))83 b Fr(\001)17 b(\001)g(\001)111 b Fm(b)-68 b Fu(')p Fw(\()p Fu(b)2432 746 y Fq(?)2432 806 y(t)2472 781 y Fw(\))2510 703 y Fm(\001)2572 523 y(0)2572 699 y(B)2572 758 y(B)2572 822 y(@)2672 585 y(b)g Fu(')p Fw(\()p Fu(b)2802 600 y Fp(1)2842 585 y Fw(\))2672 701 y Fm(b)g Fu(')p Fw(\()p Fu(b)2802 716 y Fp(2)2842 701 y Fw(\))2756 796 y(.)2756 829 y(.)2756 862 y(.)2677 978 y Fm(b)g Fu(')p Fw(\()p Fu(b)2807 993 y Fq(t)2837 978 y Fw(\))2880 523 y Fm(1)2880 699 y(C)2880 758 y(C)2880 822 y(A)1277 1330 y Fw(=)1381 1249 y Fm(\000)1426 1328 y Fu(b)1467 1292 y Fq(?)1467 1353 y Fp(1)1590 1328 y Fu(b)1631 1292 y Fq(?)1631 1353 y Fp(2)1754 1328 y Fr(\001)17 b(\001)g(\001)98 b Fu(b)2011 1292 y Fq(?)2011 1353 y(t)2051 1249 y Fm(\001)2113 1070 y(0)2113 1246 y(B)2113 1305 y(B)2113 1369 y(@)2200 1132 y Fu(b)2241 1147 y Fp(1)2200 1248 y Fu(b)2241 1263 y Fp(2)2227 1343 y Fw(.)2227 1376 y(.)2227 1409 y(.)2205 1525 y Fu(b)2246 1540 y Fq(t)2281 1070 y Fm(1)2281 1246 y(C)2281 1305 y(C)2281 1369 y(A)2396 1330 y Fw(=)2500 1236 y Fm(X)2555 1445 y Fq(q)2660 1330 y Fu(b)2701 1289 y Fq(?)2701 1355 y(q)2741 1330 y Fu(b)2782 1345 y Fq(q)2821 1330 y Fu(:)194 1721 y Fw(The)25 b(condition)d(\()843 1721 y SDict begin H.S end 843 1721 a Fw(5.13)1016 1657 y SDict begin H.R end 1016 1657 a 1016 1721 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.13) cvn H.B /ANN pdfmark end 1016 1721 a Fw(\))i(is)f(then)h(clearly)f(satis\014ed;)k(the)d (construction)g(also)e(mak)m(es)j(clear)e(that)94 1852 y(the)35 b(automorphism)44 b Fm(b)-67 b Fu(')34 b Fw(of)f Fu(R)q Fr(hh)1296 1827 y Fm(b)1274 1852 y Fu(A)p Fr(ii)g Fw(preserv)m(es)k(the)d(subalgebra)g Fu(R)q Fr(hh)2712 1827 y Fm(e)2690 1852 y Fu(A)p Fr(ii)p Fw(.)47 b(As)35 b(a)e(consequence)94 1982 y(of)k(Prop)s(osition)740 1982 y SDict begin H.S end 740 1982 a Fw(2.5)865 1918 y SDict begin H.R end 865 1918 a 865 1982 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.2.5) cvn H.B /ANN pdfmark end 865 1982 a Fw(,)51 b Fm(b)-67 b Fu(')37 b Fw(restricts)h(to)f(an)g (automorphism)f(of)h Fu(R)q Fr(hh)2618 1957 y Fm(e)2596 1982 y Fu(A)o Fr(ii)p Fw(,)i(v)m(erifying)e(\()3256 1982 y SDict begin H.S end 3256 1982 a Fw(5.12)3429 1918 y SDict begin H.R end 3429 1918 a 3429 1982 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.12) cvn H.B /ANN pdfmark end 3429 1982 a Fw(\))g(and)94 2098 y(completing)31 b(the)i(pro)s(ofs)f (of)g(Lemma)1514 2098 y SDict begin H.S end 1514 2098 a Fw(5.3)1638 2034 y SDict begin H.R end 1638 2034 a 1638 2098 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.5.3) cvn H.B /ANN pdfmark end 1638 2098 a 33 w Fw(and)g(Theorem)2272 2098 y SDict begin H.S end 2272 2098 a Fw(5.2)2397 2034 y SDict begin H.R end 2397 2034 a 2397 2098 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.5.2) cvn H.B /ANN pdfmark end 2397 2098 a Fw(.)1160 b Fh(\003)194 2319 y Fw(Note)28 b(that)g(ev)m(en)i(if)d(a)h(QP)g(\()p Fu(A;)17 b(S)6 b Fw(\))27 b(is)h(assumed)h(to)e(b)s(e)i(reduced,)h(the)f(QP)j Fm(e)-59 b Fu(\026)2971 2334 y Fq(k)3013 2319 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\))27 b(=)h(\()3463 2294 y Fm(e)3441 2319 y Fu(A;)3571 2294 y Fm(e)3558 2319 y Fu(S)5 b Fw(\))94 2450 y(is)48 b(not)f(necessarily)i(reduced)g(b)s(ecause)g(the)f(comp)s (onen)m(t)g([)p Fu(S)6 b Fw(])2469 2413 y Fp(\(2\))2617 2450 y Fr(2)2759 2424 y Fm(e)2737 2450 y Fu(A)2810 2413 y Fp(2)2897 2450 y Fw(ma)m(y)47 b(b)s(e)h(non-zero.)94 2566 y(Com)m(bining)31 b(Theorems)1041 2566 y SDict begin H.S end 1041 2566 a Fw(4.6)1165 2502 y SDict begin H.R end 1165 2502 a 1165 2566 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.6) cvn H.B /ANN pdfmark end 1165 2566 a 33 w Fw(and)1388 2566 y SDict begin H.S end 1388 2566 a Fw(5.2)1512 2502 y SDict begin H.R end 1512 2502 a 1512 2566 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.5.2) cvn H.B /ANN pdfmark end 1512 2566 a Fw(,)i(w)m(e)g(obtain)f(the)h (follo)m(wing)d(corollary)-8 b(.)94 2585 y SDict begin H.S end 94 2585 a 94 2585 a SDict begin 14 H.A end 94 2585 a 94 2585 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.5.4) cvn H.B /DEST pdfmark end 94 2585 a 177 x FA(Corollary)45 b(5.4.)g Fv(Supp)-5 b(ose)40 b(a)h(QP)f Fw(\()p Fu(A;)17 b(S)6 b Fw(\))40 b Fv(satis\014es)g Fw(\()2178 2762 y SDict begin H.S end 2178 2762 a Fw(5.1)2303 2697 y SDict begin H.R end 2303 2697 a 2303 2762 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.1) cvn H.B /ANN pdfmark end 2303 2762 a Fw(\))h Fv(and)f Fw(\()2615 2762 y SDict begin H.S end 2615 2762 a Fw(5.2)2739 2697 y SDict begin H.R end 2739 2697 a 2739 2762 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.2) cvn H.B /ANN pdfmark end 2739 2762 a Fw(\))p Fv(,)i(and)e(let)46 b Fm(e)-60 b Fu(\026)3246 2777 y Fq(k)3289 2762 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\))38 b(=)94 2892 y(\()155 2867 y Fm(e)132 2892 y Fu(A)q(;)263 2867 y Fm(e)250 2892 y Fu(S)5 b Fw(\))p Fv(.)45 b(L)-5 b(et)35 b Fw(\()p 634 2812 74 4 v Fu(A;)p 751 2812 66 4 v 17 w(S)5 b Fw(\))35 b Fv(b)-5 b(e)35 b(a)g(r)-5 b(e)g(duc)g(e)g(d) 34 b(QP)h(such)f(that)94 3101 y Fw(\(5.16\))1257 3101 y SDict begin H.S end 1257 3101 a 1257 3101 a SDict begin 14 H.A end 1257 3101 a 1257 3101 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.16) cvn H.B /DEST pdfmark end 1257 3101 a Fw(\()1318 3076 y Fm(e)1295 3101 y Fu(A;)1425 3076 y Fm(e)1412 3101 y Fu(S)6 b Fw(\))1543 3073 y Fr(\030)1544 3105 y Fw(=)1649 3101 y(\()1709 3076 y Fm(e)1687 3101 y Fu(A)1760 3116 y Fp(triv)1877 3101 y Fu(;)1934 3076 y Fm(e)1921 3101 y Fu(S)1987 3060 y Fp(\(2\))2080 3101 y Fw(\))22 b Fr(\010)h Fw(\()p 2278 3021 74 4 v Fu(A;)p 2395 3021 66 4 v 17 w(S)6 b Fw(\))94 3300 y Fv(\(se)-5 b(e)44 b Fw(\()341 3300 y SDict begin H.S end 341 3300 a Fw(4.5)465 3236 y SDict begin H.R end 465 3236 a 465 3300 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.4.5) cvn H.B /ANN pdfmark end 465 3300 a Fw(\))p Fv(\).)72 b(Then)44 b(the)g(right-e)-5 b(quivalenc)g(e)42 b(class)h(of)h Fw(\()p 2232 3220 74 4 v Fu(A;)p 2349 3220 66 4 v 17 w(S)6 b Fw(\))44 b Fv(is)f(determine)-5 b(d)43 b(by)i(the)f(right-)94 3417 y(e)-5 b(quivalenc)g(e)34 b(class)g(of)h Fw(\()p Fu(A;)17 b(S)6 b Fw(\))p Fv(.)94 3521 y SDict begin H.S end 94 3521 a 94 3521 a SDict begin 14 H.A end 94 3521 a 94 3521 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.5.5) cvn H.B /DEST pdfmark end 94 3521 a 92 x FA(De\014nition)43 b(5.5.)h Fw(In)38 b(the)g(situation)f(of)g(Corollary)2107 3613 y SDict begin H.S end 2107 3613 a Fw(5.4)2232 3548 y SDict begin H.R end 2232 3548 a 2232 3613 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.5.4) cvn H.B /ANN pdfmark end 2232 3613 a Fw(,)i(w)m(e)g(use)f(the)h(notation)d Fu(\026)3248 3628 y Fq(k)3290 3613 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\))37 b(=)94 3734 y(\()p 132 3654 74 4 v Fu(A)q(;)p 250 3654 66 4 v 17 w(S)5 b Fw(\))32 b(and)h(call)e(the)i(corresp)s (ondence)i(\()p Fu(A;)17 b(S)6 b Fw(\))27 b Fr(7!)g Fu(\026)2075 3749 y Fq(k)2118 3734 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\))32 b(the)h Fv(mutation)h(at)i(vertex)44 b Fu(k)s Fw(.)194 3930 y(Note)f(that)g(if)e(a)i(QP)g(\()p Fu(A;)17 b(S)6 b Fw(\))42 b(satis\014es)i(\()1747 3930 y SDict begin H.S end 1747 3930 a Fw(5.1)1872 3865 y SDict begin H.R end 1872 3865 a 1872 3930 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.1) cvn H.B /ANN pdfmark end 1872 3930 a Fw(\))e(then)i(the)f(same)g(is)f(true)h(for)k Fm(e)-59 b Fu(\026)3161 3945 y Fq(k)3203 3930 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\))42 b(and)94 4046 y(for)k Fu(\026)316 4061 y Fq(k)358 4046 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\).)83 b(Th)m(us,)51 b(the)c(m)m(utation)d Fu(\026)1692 4061 y Fq(k)1780 4046 y Fw(is)i(a)f(w)m(ell-de\014ned)i(transformation)c(on) j(the)h(set)94 4162 y(of)42 b(righ)m(t-equiv)-5 b(alence)40 b(classes)j(of)e(reduced)i(QPs)f(satisfying)f(\()2490 4162 y SDict begin H.S end 2490 4162 a Fw(5.1)2615 4098 y SDict begin H.R end 2615 4098 a 2615 4162 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.1) cvn H.B /ANN pdfmark end 2615 4162 a Fw(\).)70 b(\(With)41 b(some)g(abuse)i(of)94 4278 y(notation,)32 b(w)m(e)i(sometimes)e(denote)i(a)f(righ)m(t-equiv) -5 b(alence)32 b(class)h(b)m(y)h(the)f(same)g(sym)m(b)s(ol)f(as)i(an)m (y)94 4395 y(of)f(its)f(represen)m(tativ)m(es.\))94 4419 y SDict begin H.S end 94 4419 a 94 4419 a SDict begin 14 H.A end 94 4419 a 94 4419 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.5.6) cvn H.B /DEST pdfmark end 94 4419 a 171 x FA(Example)h(5.6.)39 b Fw(Consider)29 b(the)g(quiv)m(er)h Fu(Q)f Fw(with)g(v)m(ertices)h Fr(f)p Fw(1)p Fu(;)17 b Fw(2)p Fu(;)g Fw(3)p Fu(;)g Fw(4)p Fr(g)27 b Fw(and)i(arro)m(ws)g Fu(a)f Fw(:)g(1)f Fr(!)h Fw(2,)94 4707 y Fu(b)h Fw(:)e(2)h Fr(!)f Fw(3,)33 b Fu(c)27 b Fw(:)h(3)f Fr(!)h Fw(4)k(and)h Fu(d)27 b Fw(:)h(4)f Fr(!)h Fw(1:)1704 4931 y(4)1663 5083 y Fq(d)1728 5160 y Fk(\017)p Fj(\017)p 1726 5160 4 200 v 2010 4931 a Fw(3)1866 4870 y Fq(c)1781 4900 y Fk(o)p Fj(o)p 1781 4901 200 4 v 1704 5253 a Fw(1)1862 5279 y Fq(a)1981 5221 y Fk(/)p Fj(/)p 1781 5223 V 2010 5253 a Fw(2)2063 5083 y Fq(b)2034 4961 y Fk(O)p Fj(O)p 2032 5160 4 200 v 94 5464 a Fw(Let)33 b Fu(S)h Fw(=)28 b Fu(dcba)p Fw(.)45 b(Let)33 b(us)g(p)s(erform)f(the)h(m)m(utation)e (at)i(v)m(ertex)i(2.)43 b(The)34 b(arro)m(w)f Fu(a)g Fw(is)f(replaced)h(b)m(y)94 5580 y Fu(e)k Fw(:=)g Fu(a)367 5544 y Fq(?)443 5580 y Fw(:)g(2)f Fr(!)g Fw(1,)j(and)f Fu(b)g Fw(is)g(replaced)g(b)m(y)g Fu(f)48 b Fw(:=)36 b Fu(b)2027 5544 y Fq(?)2104 5580 y Fw(:)g(3)h Fr(!)f Fw(2.)59 b(W)-8 b(e)38 b(also)f(ha)m(v)m(e)i(a)f(new)g(arro)m(w)94 5696 y Fu(g)g Fw(:=)c([)p Fu(ba)p Fw(])h(:)f(1)g Fr(!)f Fw(3.)55 b(So)40 b Fm(e)-59 b Fu(\026)1103 5711 y Fp(2)1142 5696 y Fw(\()p Fu(A)p Fw(\))36 b(corresp)s(onds)i(to)e(the)h(quiv)m(er) g(with)f(v)m(ertices)h Fr(f)p Fw(1)p Fu(;)17 b Fw(2)p Fu(;)g Fw(3)p Fu(;)g Fw(4)p Fr(g)34 b Fw(and)p eop end end %%Page: 24 24 TeXDict begin HPSdict begin 24 23 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.24) cvn H.B /DEST pdfmark end 94 218 a Fs(24)492 b(HARM)25 b(DERKSEN,)f(JERZY)i(WEYMAN,)f(AND)f(ANDREI)g(ZELEVINSKY)94 417 y Fw(arro)m(ws)34 b Fu(c;)17 b(d;)g(e;)g(f)5 b(;)17 b(g)t Fw(:)1509 589 y(4)1468 870 y Fq(d)1533 1075 y Fk(\017)p Fj(\017)p 1532 1075 4 457 v 2072 589 a Fw(3)1800 528 y Fq(c)1587 557 y Fk(o)p Fj(o)p 1587 559 457 4 v 2126 864 a Fq(f)7 b Fp(=)p Fq(b)2252 840 y Fg(?)2097 1075 y Fk(\017)p Fj(\017)p 2095 1075 4 457 v 1509 1168 a Fw(1)1588 803 y Fq(g)r Fp(=[)p Fq(ba)p Fp(])2043 612 y Fk(?)p Fj(?)2014 642 y Fi(\000)1986 671 y(\000)1957 701 y(\000)1929 730 y(\000)1900 759 y(\000)1872 788 y(\000)1843 818 y(\000)1815 847 y(\000)1786 876 y(\000)1758 906 y(\000)1729 935 y(\000)1701 964 y(\000)1672 993 y(\000)1644 1023 y(\000)1615 1052 y(\000)1587 1081 y(\000)2072 1168 y Fw(2)1734 1212 y Fq(e)p Fp(=)p Fq(a)1859 1188 y Fg(?)1587 1136 y Fk(o)p Fj(o)p 1587 1138 457 4 v 94 1393 a Fw(The)34 b(p)s(oten)m(tial)h Fm(e)-59 b Fu(\026)766 1408 y Fp(2)805 1393 y Fw(\()p Fu(S)6 b Fw(\))27 b(=)1091 1367 y Fm(e)1078 1393 y Fu(S)38 b Fw(is)32 b(giv)m(en)h(b)m(y)1571 1564 y Fm(e)1557 1590 y Fu(S)h Fw(=)27 b Fu(dcg)e Fw(+)d Fu(g)t(ef)11 b Fw(;)94 1772 y(th)m(us,)39 b Fm(e)-60 b Fu(\026)395 1787 y Fp(2)434 1772 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\))32 b(is)g(reduced,)j(and)d(w)m(e) i(ha)m(v)m(e)k Fm(e)-59 b Fu(\026)1831 1787 y Fp(2)1870 1772 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\))27 b(=)h Fu(\026)2319 1787 y Fp(2)2358 1772 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\).)194 1903 y(Note)29 b(that)648 1877 y Fm(e)634 1903 y Fu(S)35 b Fw(do)s(es)30 b(not)e(satisfy)i(condition)d(\()1879 1903 y SDict begin H.S end 1879 1903 a Fw(5.2)2004 1838 y SDict begin H.R end 2004 1838 a 2004 1903 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.2) cvn H.B /ANN pdfmark end 2004 1903 a Fw(\))i(with)f(resp)s(ect)j(to)d(v)m(ertex)j Fu(k)g Fw(=)d(3)g(since)i(the)94 2033 y(path)40 b Fu(g)t(ef)49 b Fw(starts)39 b(and)g(ends)i(at)d(3.)63 b(But)39 b(w)m(e)h(can)f (\014x)h(this)f(condition)e(b)m(y)j(replacing)3380 2008 y Fm(e)3367 2033 y Fu(S)45 b Fw(with)94 2158 y(a)36 b(cyclically)e (equiv)-5 b(alen)m(t)36 b(p)s(oten)m(tial,)f(sa)m(y)i Fu(S)1753 2122 y Fn(0)1810 2158 y Fw(=)c Fu(dcg)28 b Fw(+)c Fu(ef)11 b(g)t Fw(.)53 b(No)m(w)36 b(let)g(us)g(m)m(utate)g(\() 3317 2133 y Fm(e)3295 2158 y Fu(A;)17 b(S)3478 2122 y Fn(0)3501 2158 y Fw(\))36 b(at)94 2274 y(v)m(ertex)i(3.)53 b(The)37 b(arro)m(ws)f Fu(c;)17 b(f)5 b(;)17 b(g)39 b Fw(are)d(replaced)g(b)m(y)g Fu(c)2044 2238 y Fq(?)2117 2274 y Fw(:)d(4)g Fr(!)g Fw(3,)j Fu(f)2563 2238 y Fq(?)2635 2274 y Fw(:)e(2)f Fr(!)f Fw(3)k(and)g Fu(g)3239 2238 y Fq(?)3311 2274 y Fw(:)d(3)g Fr(!)g Fw(1,)94 2399 y(resp)s(ectiv)m (ely)-8 b(.)46 b(W)-8 b(e)34 b(also)e(add)h(new)h(arro)m(ws)g([)p Fu(cg)t Fw(])28 b(:)g(1)g Fr(!)h Fw(4)j(and)h([)p Fu(f)11 b(g)t Fw(])28 b(:)h(1)f Fr(!)g Fw(2.)45 b(Th)m(us,)39 b Fm(e)-59 b Fu(\026)3341 2414 y Fp(3)3380 2399 y Fw(\()3440 2374 y Fm(e)3418 2399 y Fu(A;)17 b(S)3601 2363 y Fn(0)3623 2399 y Fw(\))94 2515 y(has)33 b(arro)m(ws)h Fr(f)p Fu(d;)17 b(e;)g(c)857 2479 y Fq(?)895 2515 y Fu(;)g(f)998 2479 y Fq(?)1037 2515 y Fu(;)g(g)1132 2479 y Fq(?)1170 2515 y Fu(;)g Fw([)p Fu(cg)t Fw(])p Fu(;)g Fw([)p Fu(f)11 b(g)t Fw(])p Fr(g)p Fw(:)1562 2751 y(4)1835 2691 y Fq(c)1866 2667 y Fg(?)2097 2720 y Fk(/)p Fj(/)p 1640 2721 457 4 v 1500 3032 a Fq(d)1565 3237 y Fk(\017)p Fj(\017)p 1563 3237 4 457 v 2126 2751 a Fw(3)1767 2969 y Fq(g)1803 2945 y Fg(?)1640 3244 y Fk(\177)p Fj(\177)p Fi(\000)1668 3214 y(\000)1697 3185 y(\000)1725 3156 y(\000)1754 3126 y(\000)1782 3097 y(\000)1811 3068 y(\000)1839 3039 y(\000)1868 3009 y(\000)1896 2980 y(\000)1925 2951 y(\000)1953 2921 y(\000)1982 2892 y(\000)2011 2863 y(\000)2039 2834 y(\000)2068 2804 y(\000)1562 3331 y Fw(1)1637 3026 y Fp([)p Fq(cg)r Fp(])1608 2781 y Fk(O)p Fj(O)p 1606 3237 V 1810 3231 a Fp([)p Fq(f)c(g)r Fp(])2097 3277 y Fk(/)p Fj(/)p 1640 3279 457 4 v 2126 3331 a Fw(2)2179 3026 y Fq(f)2220 3002 y Fg(?)2150 2781 y Fk(O)p Fj(O)p 2148 3237 4 457 v 1852 3378 a Fq(e)1640 3320 y Fk(o)p Fj(o)p 1640 3322 457 4 v 94 3556 a Fw(The)34 b(p)s(oten)m(tial)h Fm(e)-59 b Fu(\026)766 3571 y Fp(3)805 3556 y Fw(\()p Fu(S)909 3520 y Fn(0)932 3556 y Fw(\))32 b(is)h(giv)m(en)f(b)m(y)950 3739 y Fu(\026)1009 3754 y Fp(3)1048 3739 y Fw(\()p Fu(S)1152 3698 y Fn(0)1175 3739 y Fw(\))c(=)f Fu(d)p Fw([)p Fu(cg)t Fw(])21 b(+)i Fu(e)p Fw([)p Fu(f)11 b(g)t Fw(])21 b(+)h([)p Fu(f)11 b(g)t Fw(])p Fu(g)2205 3698 y Fq(?)2243 3739 y Fu(f)2302 3698 y Fq(?)2363 3739 y Fw(+)22 b([)p Fu(cg)t Fw(])p Fu(g)2659 3698 y Fq(?)2697 3739 y Fu(c)2739 3698 y Fq(?)2779 3739 y Fu(:)94 3936 y Fw(It)43 b(is)f(not)g(reduced,)47 b(so)42 b(to)g(obtain)g(the)g(reduced)i(QP)f Fu(\026)2270 3951 y Fp(3)2309 3936 y Fw(\()2370 3911 y Fm(e)2347 3936 y Fu(A;)17 b(S)2530 3900 y Fn(0)2553 3936 y Fw(\),)45 b(w)m(e)e(need)h(to)e(remo)m(v)m(e)h(the)94 4066 y(trivial)30 b(part)j(of)j Fm(e)-59 b Fu(\026)766 4081 y Fp(3)805 4066 y Fw(\()865 4041 y Fm(e)843 4066 y Fu(A;)17 b(S)1026 4030 y Fn(0)1049 4066 y Fw(\).)43 b(The)34 b(resulting)d(quiv)m(er)j (is)e(as)h(follo)m(ws:)1531 4303 y(4)1804 4242 y Fq(c)1835 4218 y Fg(?)2065 4271 y Fk(/)p Fj(/)p 1609 4272 V 2094 4303 a Fw(3)1736 4520 y Fq(g)1772 4496 y Fg(?)1609 4795 y Fk(\177)p Fj(\177)p Fi(\000)1637 4766 y(\000)1666 4736 y(\000)1694 4707 y(\000)1723 4678 y(\000)1751 4648 y(\000)1780 4619 y(\000)1808 4590 y(\000)1837 4561 y(\000)1865 4531 y(\000)1894 4502 y(\000)1922 4473 y(\000)1951 4443 y(\000)1979 4414 y(\000)2008 4385 y(\000)2036 4356 y(\000)1531 4882 y Fw(1)514 b(2)2148 4577 y Fq(f)2189 4554 y Fg(?)2119 4332 y Fk(O)p Fj(O)p 2117 4789 4 457 v 94 5061 a Fw(Since)33 b(it)f(is)g(acyclic)g(\(that)g(is,)g(has)h(no)g(orien)m(ted)f (cycles\),)i(the)f(corresp)s(onding)g(p)s(oten)m(tial)e(is)h(0.)194 5250 y(Our)h(next)g(result)g(is)f(that)g(ev)m(ery)j(m)m(utation)c(is)h (an)g(in)m(v)m(olution.)94 5269 y SDict begin H.S end 94 5269 a 94 5269 a SDict begin 14 H.A end 94 5269 a 94 5269 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.5.7) cvn H.B /DEST pdfmark end 94 5269 a 170 x FA(Theorem)44 b(5.7.)h Fv(The)40 b(c)-5 b(orr)g(esp)g(ondenc)g(e)38 b Fu(\026)1734 5454 y Fq(k)1814 5439 y Fw(:)g(\()p Fu(A;)17 b(S)6 b Fw(\))37 b Fr(!)g Fw(\()p 2350 5359 74 4 v Fu(A;)p 2467 5359 66 4 v 17 w(S)6 b Fw(\))40 b Fv(acts)g(as)g(an)g(involution)g(on) 94 5555 y(the)d(set)f(of)g(right-e)-5 b(quivalenc)g(e)35 b(classes)g(of)h(r)-5 b(e)g(duc)g(e)g(d)35 b(QPs)h(satisfying)g Fw(\()2736 5555 y SDict begin H.S end 2736 5555 a Fw(5.1)2861 5491 y SDict begin H.R end 2861 5491 a 2861 5555 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.1) cvn H.B /ANN pdfmark end 2861 5555 a Fw(\))p Fv(,)g(that)g(is,)h Fu(\026)3361 5519 y Fp(2)3361 5581 y Fq(k)3403 5555 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\))94 5671 y Fv(is)35 b(right-e)-5 b(quivalent)34 b(to)h Fw(\()p Fu(A;)17 b(S)6 b Fw(\))p Fv(.)p eop end end %%Page: 25 25 TeXDict begin HPSdict begin 25 24 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.25) cvn H.B /DEST pdfmark end 94 218 a 1179 w Fs(QUIVERS)31 b(WITH)i(POTENTIALS)g(I)1102 b(25)94 431 y Fv(Pr)-5 b(o)g(of.)42 b Fw(Let)32 b(\()p Fu(A;)17 b(S)6 b Fw(\))32 b(b)s(e)h(a)f(reduced)i(QP)e(satisfying)g(\()2082 431 y SDict begin H.S end 2082 431 a Fw(5.1)2206 367 y SDict begin H.R end 2206 367 a 2206 431 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.1) cvn H.B /ANN pdfmark end 2206 431 a Fw(\))h(and)f(\()2504 431 y SDict begin H.S end 2504 431 a Fw(5.2)2629 367 y SDict begin H.R end 2629 367 a 2629 431 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.2) cvn H.B /ANN pdfmark end 2629 431 a Fw(\).)43 b(Let)37 b Fm(e)-59 b Fu(\026)2971 446 y Fq(k)3013 431 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\))27 b(=)h(\()3463 406 y Fm(e)3441 431 y Fu(A;)3571 406 y Fm(e)3558 431 y Fu(S)5 b Fw(\))94 591 y(and)35 b Fm(e)-59 b Fu(\026)341 554 y Fp(2)341 616 y Fq(k)383 591 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\))27 b(=)32 b Fm(e)-59 b Fu(\026)832 606 y Fq(k)874 591 y Fw(\()935 565 y Fm(e)912 591 y Fu(A;)1042 565 y Fm(e)1029 591 y Fu(S)6 b Fw(\))28 b(=)f(\()1311 536 y Fm(e)1324 565 y(e)1302 591 y Fu(A;)1424 536 y Fm(e)1432 565 y(e)1419 591 y Fu(S)6 b Fw(\).)42 b(In)31 b(view)g(of)e(Theorem)2453 591 y SDict begin H.S end 2453 591 a Fw(4.6)2578 526 y SDict begin H.R end 2578 526 a 2578 591 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.6) cvn H.B /ANN pdfmark end 2578 591 a 30 w Fw(and)h(Prop)s(osition)3318 591 y SDict begin H.S end 3318 591 a Fw(5.1)3443 526 y SDict begin H.R end 3443 526 a 3443 591 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.5.1) cvn H.B /ANN pdfmark end 3443 591 a Fw(,)h(it)e(is)94 707 y(enough)k(to)g(sho)m(w)g(that)94 910 y(\(5.17\))445 910 y SDict begin H.S end 445 910 a 445 910 a SDict begin 14 H.A end 445 910 a 445 910 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.17) cvn H.B /DEST pdfmark end 445 910 a Fw(\()492 855 y Fm(e)505 884 y(e)483 910 y Fu(A;)605 855 y Fm(e)613 884 y(e)600 910 y Fu(S)5 b Fw(\))33 b(is)f(righ)m(t-equiv)-5 b(alen)m(t)31 b(to)h(\()p Fu(A;)17 b(S)6 b Fw(\))22 b Fr(\010)g Fw(\()p Fu(C)r(;)17 b(T)d Fw(\),)32 b(where)i(\()p Fu(C)r(;)17 b(T)d Fw(\))31 b(is)h(a)g(trivial)e(QP)-8 b(.)94 1075 y(Using)39 b(\()413 1075 y SDict begin H.S end 413 1075 a Fw(5.4)538 1011 y SDict begin H.R end 538 1011 a 538 1075 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.4) cvn H.B /ANN pdfmark end 538 1075 a Fw(\))g(t)m(wice,)i(and)f(iden)m(tifying)d(\()p Fu(e)1673 1090 y Fq(k)1716 1075 y Fu(A)p Fw(\))1827 1039 y Fq(?)1906 1075 y Fw(with)i Fu(A)2208 1039 y Fq(?)2247 1075 y Fu(e)2292 1090 y Fq(k)2335 1075 y Fw(,)i(and)e(\()p Fu(Ae)2755 1090 y Fq(k)2798 1075 y Fw(\))2836 1039 y Fq(?)2915 1075 y Fw(with)g Fu(e)3189 1090 y Fq(k)3232 1075 y Fu(A)3305 1039 y Fq(?)3344 1075 y Fw(,)i(where)94 1191 y Fu(A)167 1155 y Fq(?)240 1191 y Fw(is)32 b(the)h(dual)e Fu(R)q Fw(-bimo)s(dule)f(of)j Fu(A)p Fw(,)f(w)m(e)i(conclude)f(that)94 1394 y(\(5.18\))1330 1394 y SDict begin H.S end 1330 1394 a 1330 1394 a SDict begin 14 H.A end 1330 1394 a 1330 1394 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.18) cvn H.B /DEST pdfmark end 1330 1394 a 1339 1340 a Fm(e)1353 1369 y(e)1330 1394 y Fu(A)28 b Fw(=)g Fu(A)22 b Fr(\010)h Fu(Ae)1848 1409 y Fq(k)1891 1394 y Fu(A)f Fr(\010)g Fu(A)2158 1353 y Fq(?)2198 1394 y Fu(e)2243 1409 y Fq(k)2286 1394 y Fu(A)2359 1353 y Fq(?)2398 1394 y Fu(:)94 1602 y Fw(F)-8 b(urthermore,)34 b(the)h(basis)f(of)f(arro)m(ws)i(in)1649 1548 y Fm(e)1662 1577 y(e)1640 1602 y Fu(A)f Fw(consists)g(of)g(the)g (original)d(set)k(of)e(arro)m(ws)i Fu(Q)3400 1617 y Fp(1)3473 1602 y Fw(in)e Fu(A)94 1719 y Fw(together)41 b(with)f(the)h(arro)m(ws)g ([)p Fu(ba)p Fw(])h Fr(2)f Fu(Ae)1626 1734 y Fq(k)1669 1719 y Fu(A)g Fw(and)g([)p Fu(a)2059 1683 y Fq(?)2098 1719 y Fu(b)2139 1683 y Fq(?)2179 1719 y Fw(])g Fr(2)h Fu(A)2428 1683 y Fq(?)2467 1719 y Fu(e)2512 1734 y Fq(k)2555 1719 y Fu(A)2628 1683 y Fq(?)2708 1719 y Fw(for)e Fu(a)h Fr(2)h Fu(Q)3142 1734 y Fp(1)3209 1719 y Fr(\\)28 b Fu(e)3348 1734 y Fq(k)3391 1719 y Fu(A)40 b Fw(and)94 1835 y Fu(b)29 b Fr(2)f Fu(Q)335 1850 y Fp(1)397 1835 y Fr(\\)22 b Fu(Ae)603 1850 y Fq(k)646 1835 y Fw(.)194 1989 y(Remem)m(b)s(ering)31 b(\()851 1989 y SDict begin H.S end 851 1989 a Fw(5.8)976 1925 y SDict begin H.R end 976 1925 a 976 1989 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.8) cvn H.B /ANN pdfmark end 976 1989 a Fw(\))h(and)h(\()1274 1989 y SDict begin H.S end 1274 1989 a Fw(5.9)1399 1925 y SDict begin H.R end 1399 1925 a 1399 1989 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.9) cvn H.B /ANN pdfmark end 1399 1989 a Fw(\),)f(w)m(e)i(see)f(that)g(the)g(p)s(oten)m(tial)2594 1935 y Fm(e)2602 1964 y(e)2589 1989 y Fu(S)38 b Fw(is)32 b(giv)m(en)h(b)m(y)94 2197 y(\(5.19\))420 2197 y SDict begin H.S end 420 2197 a 420 2197 a SDict begin 14 H.A end 420 2197 a 420 2197 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.19) cvn H.B /DEST pdfmark end 420 2197 a 425 2143 a Fm(e)433 2172 y(e)420 2197 y Fu(S)g Fw(=)28 b([[)p Fu(S)6 b Fw(]])22 b(+)g([\001)1019 2212 y Fq(k)1062 2197 y Fw(\()p Fu(A)p Fw(\)])g(+)g(\001)1439 2212 y Fq(k)1482 2197 y Fw(\()1543 2172 y Fm(e)1520 2197 y Fu(A)p Fw(\))28 b(=)g([)p Fu(S)6 b Fw(])22 b(+)2265 2103 y Fm(X)2003 2318 y Fq(a;b)p Fn(2)p Fq(Q)2193 2327 y Fo(1)2227 2318 y Fp(:)32 b Fq(h)p Fp(\()p Fq(a)p Fp(\)=)p Fq(t)p Fp(\()p Fq(b)p Fp(\)=)p Fq(k)2671 2197 y Fw(\([)p Fu(ba)p Fw(][)p Fu(a)2933 2156 y Fq(?)2974 2197 y Fu(b)3015 2156 y Fq(?)3055 2197 y Fw(])22 b(+)g([)p Fu(a)3280 2156 y Fq(?)3320 2197 y Fu(b)3361 2156 y Fq(?)3401 2197 y Fw(])p Fu(ba)p Fw(\))p Fu(;)94 2471 y Fw(hence)34 b(it)e(is)g(cyclically)f(equiv)-5 b(alen)m(t)32 b(to)94 2653 y(\(5.20\))978 2653 y SDict begin H.S end 978 2653 a 978 2653 a SDict begin 14 H.A end 978 2653 a 978 2653 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.20) cvn H.B /DEST pdfmark end 978 2653 a Fu(S)1038 2668 y Fp(1)1105 2653 y Fw(=)27 b([)p Fu(S)6 b Fw(])22 b(+)1711 2558 y Fm(X)1448 2774 y Fq(a;b)p Fn(2)p Fq(Q)1638 2783 y Fo(1)1673 2774 y Fp(:)32 b Fq(h)p Fp(\()p Fq(a)p Fp(\)=)p Fq(t)p Fp(\()p Fq(b)p Fp(\)=)p Fq(k)2117 2653 y Fw(\([)p Fu(ba)p Fw(])23 b(+)f Fu(ba)p Fw(\)[)p Fu(a)2630 2612 y Fq(?)2670 2653 y Fu(b)2711 2612 y Fq(?)2751 2653 y Fw(])94 2933 y(\(recall)h(that)h([)p Fu(S)6 b Fw(])24 b(is)f(obtained)h(b)m(y)h(substituting)e([)p Fu(a)1952 2948 y Fq(p)1992 2933 y Fu(a)2043 2948 y Fq(p)p Fp(+1)2173 2933 y Fw(])h(for)g(eac)m(h)h(factor)e Fu(a)2897 2948 y Fq(p)2937 2933 y Fu(a)2988 2948 y Fq(p)p Fp(+1)3142 2933 y Fw(with)h Fu(t)p Fw(\()p Fu(a)3480 2948 y Fq(p)3520 2933 y Fw(\))k(=)94 3049 y Fu(h)p Fw(\()p Fu(a)239 3064 y Fq(p)p Fp(+1)370 3049 y Fw(\))33 b(=)g Fu(k)39 b Fw(of)c(an)m(y)i (cyclic)e(path)h Fu(a)1495 3064 y Fp(1)1551 3049 y Fr(\001)17 b(\001)g(\001)e Fu(a)1735 3064 y Fq(d)1811 3049 y Fw(o)s(ccurring)35 b(in)h(the)g(path)g(expansion)g(of)f Fu(S)6 b Fw(\).)53 b(Let)94 3165 y(us)34 b(abbreviate)753 3330 y(\()p Fu(C)r(;)17 b(T)d Fw(\))27 b(=)g(\()p Fu(Ae)1302 3345 y Fq(k)1345 3330 y Fu(A)22 b Fr(\010)h Fu(A)1613 3289 y Fq(?)1653 3330 y Fu(e)1698 3345 y Fq(k)1740 3330 y Fu(A)1813 3289 y Fq(?)1853 3330 y Fu(;)2159 3236 y Fm(X)1897 3452 y Fq(a;b)p Fn(2)p Fq(Q)2087 3461 y Fo(1)2121 3452 y Fp(:)32 b Fq(h)p Fp(\()p Fq(a)p Fp(\)=)p Fq(t)p Fp(\()p Fq(b)p Fp(\)=)p Fq(k)2565 3330 y Fw([)p Fu(ba)p Fw(][)p Fu(a)2789 3289 y Fq(?)2830 3330 y Fu(b)2871 3289 y Fq(?)2911 3330 y Fw(]\))p Fu(:)94 3610 y Fw(This)i(is)f(a)g(trivial)e(QP)j(\(cf.)46 b(Prop)s(osition)1676 3610 y SDict begin H.S end 1676 3610 a Fw(4.4)1800 3546 y SDict begin H.R end 1800 3546 a 1800 3610 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.4) cvn H.B /ANN pdfmark end 1800 3610 a Fw(\);)34 b(therefore)g(to)f(pro)m(v)m(e)i(Theorem)3106 3610 y SDict begin H.S end 3106 3610 a Fw(5.7)3230 3546 y SDict begin H.R end 3230 3546 a 3230 3610 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.5.7) cvn H.B /ANN pdfmark end 3230 3610 a 34 w Fw(it)d(su\016ces)94 3770 y(to)27 b(sho)m(w)h(that)f(the)g(QP)g(\()1028 3715 y Fm(e)1042 3744 y(e)1019 3770 y Fu(A;)17 b(S)1196 3785 y Fp(1)1235 3770 y Fw(\))27 b(giv)m(en)g(b)m(y)h(\()1717 3770 y SDict begin H.S end 1717 3770 a Fw(5.18)1890 3705 y SDict begin H.R end 1890 3705 a 1890 3770 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.18) cvn H.B /ANN pdfmark end 1890 3770 a Fw(\))f(and)g(\()2177 3770 y SDict begin H.S end 2177 3770 a Fw(5.20)2351 3705 y SDict begin H.R end 2351 3705 a 2351 3770 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.20) cvn H.B /ANN pdfmark end 2351 3770 a Fw(\))f(is)h(righ)m(t-equiv) -5 b(alen)m(t)25 b(to)i(\()p Fu(A;)17 b(S)6 b Fw(\))11 b Fr(\010)94 3886 y Fw(\()p Fu(C)r(;)17 b(T)d Fw(\).)43 b(W)-8 b(e)33 b(pro)s(ceed)g(in)f(sev)m(eral)h(steps.)194 4045 y FA(Step)50 b(1:)64 b Fw(Let)44 b Fu(')864 4060 y Fp(1)946 4045 y Fw(b)s(e)g(the)f(c)m(hange)h(of)f(arro)m(ws)g (automorphism)e(of)i Fu(R)q Fr(hh)2985 3991 y Fm(e)2998 4020 y(e)2976 4045 y Fu(A)p Fr(ii)f Fw(\(see)j(De\014ni-)94 4161 y(tion)306 4161 y SDict begin H.S end 306 4161 a Fw(2.6)430 4097 y SDict begin H.R end 430 4097 a 430 4161 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.2.6) cvn H.B /ANN pdfmark end 430 4161 a Fw(\))e(m)m(ultiplying)d(eac)m(h)k(arro)m (w)f Fu(b)j Fr(2)g Fu(Q)1832 4176 y Fp(1)1901 4161 y Fr(\\)30 b Fu(Ae)2115 4176 y Fq(k)2201 4161 y Fw(b)m(y)44 b Fr(\000)p Fw(1,)i(and)d(\014xing)g(the)h(rest)f(of)g(the)94 4321 y(arro)m(ws)34 b(in)529 4267 y Fm(e)543 4296 y(e)520 4321 y Fu(A)q Fw(.)43 b(Then)34 b(the)f(p)s(oten)m(tial)e Fu(S)1559 4336 y Fp(2)1626 4321 y Fw(=)c Fu(')1793 4336 y Fp(1)1833 4321 y Fw(\()p Fu(S)1931 4336 y Fp(1)1970 4321 y Fw(\))32 b(is)h(giv)m(en)f(b)m(y)963 4508 y Fu(S)1023 4523 y Fp(2)1090 4508 y Fw(=)c([)p Fu(S)6 b Fw(])22 b(+)1696 4413 y Fm(X)1434 4629 y Fq(a;b)p Fn(2)p Fq(Q)1624 4638 y Fo(1)1659 4629 y Fp(:)32 b Fq(h)p Fp(\()p Fq(a)p Fp(\)=)p Fq(t)p Fp(\()p Fq(b)p Fp(\)=)p Fq(k)2103 4508 y Fw(\([)p Fu(ba)p Fw(])22 b Fr(\000)h Fu(ba)p Fw(\)[)p Fu(a)2617 4467 y Fq(?)2657 4508 y Fu(b)2698 4467 y Fq(?)2738 4508 y Fw(])p Fu(:)194 4839 y FA(Step)35 b(2:)43 b Fw(Let)30 b Fu(')814 4854 y Fp(2)884 4839 y Fw(b)s(e)g(the)h(unitriangular)c (automorphism)i(of)g Fu(R)q Fr(hh)2682 4785 y Fm(e)2695 4814 y(e)2673 4839 y Fu(A)p Fr(ii)h Fw(\(see)h(De\014nition)3499 4839 y SDict begin H.S end 3499 4839 a Fw(2.6)3623 4775 y SDict begin H.R end 3623 4775 a 3623 4839 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.2.6) cvn H.B /ANN pdfmark end 3623 4839 a Fw(\))94 4955 y(sending)42 b(eac)m(h)g(arro)m(w)g([) p Fu(ba)p Fw(])h Fr(2)g Fu(Ae)1383 4970 y Fq(k)1426 4955 y Fu(A)f Fw(to)f([)p Fu(ba)p Fw(])29 b(+)f Fu(ba)p Fw(,)44 b(and)e(\014xing)f(the)g(rest)h(of)f(the)h(arro)m(ws)g(in)103 5060 y Fm(e)117 5089 y(e)94 5115 y Fu(A)q Fw(.)53 b(Remem)m(b)s(ering) 34 b(the)j(de\014nition)e(of)g([)p Fu(S)6 b Fw(],)37 b(it)d(is)i(easy)h(to)e(see)i(that)f(the)g(p)s(oten)m(tial)e Fu(')3345 5130 y Fp(2)3385 5115 y Fw(\()p Fu(S)3483 5130 y Fp(2)3522 5115 y Fw(\))i(is)94 5231 y(cyclically)31 b(equiv)-5 b(alen)m(t)33 b(to)f(a)g(p)s(oten)m(tial)f(of)h(the)h(form) 916 5412 y Fu(S)976 5427 y Fp(3)1043 5412 y Fw(=)27 b Fu(S)h Fw(+)1594 5317 y Fm(X)1332 5533 y Fq(a;b)p Fn(2)p Fq(Q)1522 5542 y Fo(1)1557 5533 y Fp(:)k Fq(h)p Fp(\()p Fq(a)p Fp(\)=)p Fq(t)p Fp(\()p Fq(b)p Fp(\)=)p Fq(k)2001 5412 y Fw([)p Fu(ba)p Fw(]\([)p Fu(a)2263 5371 y Fq(?)2303 5412 y Fu(b)2344 5371 y Fq(?)2384 5412 y Fw(])22 b(+)g Fu(f)11 b Fw(\()p Fu(a;)17 b(b)p Fw(\)\))94 5696 y(for)33 b(some)f(elemen)m(ts)h Fu(f)11 b Fw(\()p Fu(a;)17 b(b)p Fw(\))27 b Fr(2)i Fl(m)p Fw(\()p Fu(A)22 b Fr(\010)h Fu(Ae)1707 5711 y Fq(k)1750 5696 y Fu(A)p Fw(\))1861 5660 y Fp(2)1900 5696 y Fw(.)p eop end end %%Page: 26 26 TeXDict begin HPSdict begin 26 25 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.26) cvn H.B /DEST pdfmark end 94 218 a Fs(26)492 b(HARM)25 b(DERKSEN,)f(JERZY)i(WEYMAN,)f(AND)f(ANDREI)g(ZELEVINSKY)194 460 y FA(Step)35 b(3:)42 b Fw(Let)31 b Fu(')814 475 y Fp(3)883 460 y Fw(b)s(e)g(the)f(unitriangular)e(automorphism)g(of)h Fu(R)q Fr(hh)2681 406 y Fm(e)2694 435 y(e)2672 460 y Fu(A)p Fr(ii)h Fw(sending)g(eac)m(h)h(arro)m(w)94 620 y([)p Fu(a)172 583 y Fq(?)212 620 y Fu(b)253 583 y Fq(?)293 620 y Fw(])h Fr(2)g Fu(A)523 583 y Fq(?)562 620 y Fu(e)607 635 y Fq(k)650 620 y Fu(A)723 583 y Fq(?)798 620 y Fw(to)i([)p Fu(a)997 583 y Fq(?)1037 620 y Fu(b)1078 583 y Fq(?)1118 620 y Fw(])24 b Fr(\000)g Fu(f)11 b Fw(\()p Fu(a;)17 b(b)p Fw(\),)35 b(and)g(\014xing)g(the)g(rest)h(of)e(the)h(arro)m(ws)h (in)3152 565 y Fm(e)3165 594 y(e)3143 620 y Fu(A)p Fw(.)50 b(Then)36 b(w)m(e)94 736 y(ha)m(v)m(e)e Fu(')383 751 y Fp(3)423 736 y Fw(\()p Fu(S)521 751 y Fp(3)560 736 y Fw(\))28 b(=)f Fu(S)h Fw(+)22 b Fu(T)14 b Fw(.)194 895 y(Com)m(bining)32 b(these)j(three)f(steps,)i(w)m(e)e(conclude)h (that)e(the)h(QP)g(\()2617 841 y Fm(e)2630 870 y(e)2608 895 y Fu(A;)17 b(S)2785 910 y Fp(1)2824 895 y Fw(\))34 b(is)f(righ)m(t-equiv)-5 b(alen)m(t)94 1055 y(to)33 b(\()261 1000 y Fm(e)274 1029 y(e)252 1055 y Fu(A;)17 b(S)27 b Fw(+)22 b Fu(T)14 b Fw(\))28 b(=)f(\()p Fu(A;)17 b(S)6 b Fw(\))22 b Fr(\010)h Fw(\()p Fu(C)r(;)17 b(T)d Fw(\),)31 b(\014nishing)h(the)h(pro)s(of)e(of)i(Theorem)2832 1055 y SDict begin H.S end 2832 1055 a Fw(5.7)2957 990 y SDict begin H.R end 2957 990 a 2957 1055 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.5.7) cvn H.B /ANN pdfmark end 2957 1055 a Fw(.)600 b Fh(\003)94 1190 y SDict begin H.S end 94 1190 a 94 1190 a SDict begin 14 H.A end 94 1190 a 94 1190 a SDict begin [ /View [/XYZ H.V] /Dest (section.6) cvn H.B /DEST pdfmark end 94 1190 a 1162 1281 a Fw(6.)49 b Fx(Some)38 b(mut)-7 b(a)g(tion)39 b(inv)-9 b(ariants)194 1455 y Fw(In)39 b(this)f(section)g(w)m(e)h(\014x)g(a)f(v)m(ertex)i Fu(k)h Fw(and)e(study)g(the)g(e\013ect)g(of)f(the)h(m)m(utation)d Fu(\026)3304 1470 y Fq(k)3385 1455 y Fw(on)i(the)94 1571 y(Jacobian)28 b(algebra)g Fr(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\).)42 b(W)-8 b(e)29 b(will)e(use)j(the)f(follo)m(wing)d (notation:)40 b(for)28 b(an)h Fu(R)q Fw(-bimo)s(dule)d Fu(B)5 b Fw(,)94 1688 y(denote)94 1855 y(\(6.1\))1362 1855 y SDict begin H.S end 1362 1855 a 1362 1855 a SDict begin 14 H.A end 1362 1855 a 1362 1855 a SDict begin [ /View [/XYZ H.V] /Dest (equation.6.1) cvn H.B /DEST pdfmark end 1362 1855 a Fu(B)1438 1867 y Fp(^)1436 1884 y Fq(k)q(;)1496 1866 y Fp(^)1494 1884 y Fq(k)1564 1855 y Fw(=)p 1668 1800 46 4 v 28 w Fu(e)1713 1870 y Fq(k)1756 1855 y Fu(B)p 1835 1800 V 5 w(e)1880 1870 y Fq(k)1950 1855 y Fw(=)2063 1760 y Fm(M)2054 1972 y Fq(i;j)t Fn(6)p Fp(=)p Fq(k)2240 1855 y Fu(B)2314 1870 y Fq(i;j)94 2129 y Fw(\(see)33 b(\()327 2129 y SDict begin H.S end 327 2129 a Fw(5.5)452 2065 y SDict begin H.R end 452 2065 a 452 2129 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.5) cvn H.B /ANN pdfmark end 452 2129 a Fw(\)\).)43 b(Note)32 b(that)f(if)f Fu(B)37 b Fw(is)31 b(a)h(\(top)s(ological\))27 b Fu(R)q Fw(-algebra)j(then)j Fu(B)2736 2141 y Fp(^)2734 2158 y Fq(k)q(;)2794 2141 y Fp(^)2792 2158 y Fq(k)2866 2129 y Fw(is)e(a)g(\(closed\))h(subal-)94 2249 y(gebra)h(of)f Fu(B)5 b Fw(.)94 2268 y SDict begin H.S end 94 2268 a 94 2268 a SDict begin 14 H.A end 94 2268 a 94 2268 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.6.1) cvn H.B /DEST pdfmark end 94 2268 a 169 x FA(Prop)s(osition)43 b(6.1.)i Fv(Supp)-5 b(ose)39 b(a)i(QP)f Fw(\()p Fu(A;)17 b(S)6 b Fw(\))40 b Fv(satis\014es)f Fw(\()2282 2437 y SDict begin H.S end 2282 2437 a Fw(5.1)2407 2373 y SDict begin H.R end 2407 2373 a 2407 2437 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.1) cvn H.B /ANN pdfmark end 2407 2437 a Fw(\))h Fv(and)g Fw(\()2718 2437 y SDict begin H.S end 2718 2437 a Fw(5.2)2842 2373 y SDict begin H.R end 2842 2373 a 2842 2437 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.2) cvn H.B /ANN pdfmark end 2842 2437 a Fw(\))p Fv(,)i(and)e(let)g Fw(\()3350 2412 y Fm(e)3327 2437 y Fu(A;)3457 2412 y Fm(e)3444 2437 y Fu(S)6 b Fw(\))38 b(=)98 2568 y Fm(e)-59 b Fu(\026)153 2583 y Fq(k)196 2568 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\))32 b Fv(b)-5 b(e)33 b(given)f(by)h Fw(\()1020 2568 y SDict begin H.S end 1020 2568 a Fw(5.4)1145 2503 y SDict begin H.R end 1145 2503 a 1145 2568 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.4) cvn H.B /ANN pdfmark end 1145 2568 a Fw(\))g Fv(and)f Fw(\()1441 2568 y SDict begin H.S end 1441 2568 a Fw(5.8)1566 2503 y SDict begin H.R end 1566 2503 a 1566 2568 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.8) cvn H.B /ANN pdfmark end 1566 2568 a Fw(\))p Fv(.)44 b(Then)32 b(the)h Fu(R)q Fv(-algebr)-5 b(as)32 b Fr(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\))2906 2580 y Fp(^)2904 2597 y Fq(k)q(;)2964 2579 y Fp(^)2962 2597 y Fq(k)3037 2568 y Fv(and)33 b Fr(P)8 b Fw(\()3363 2542 y Fm(e)3340 2568 y Fu(A;)3470 2542 y Fm(e)3457 2568 y Fu(S)e Fw(\))3563 2580 y Fp(^)3561 2597 y Fq(k)q(;)3621 2579 y Fp(^)3619 2597 y Fq(k)94 2687 y Fv(ar)-5 b(e)35 b(isomorphic)e(to)j(e)-5 b(ach)34 b(other.)94 2867 y(Pr)-5 b(o)g(of.)42 b Fw(In)33 b(view)f(of)h(\()889 2867 y SDict begin H.S end 889 2867 a Fw(5.4)1013 2803 y SDict begin H.R end 1013 2803 a 1013 2867 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.4) cvn H.B /ANN pdfmark end 1013 2867 a Fw(\),)g(w)m(e)g(ha)m(v)m(e)94 3048 y(\(6.2\))1447 3048 y SDict begin H.S end 1447 3048 a 1447 3048 a SDict begin 14 H.A end 1447 3048 a 1447 3048 a SDict begin [ /View [/XYZ H.V] /Dest (equation.6.2) cvn H.B /DEST pdfmark end 1447 3048 a 1469 3023 a Fm(e)1447 3048 y Fu(A)1522 3060 y Fp(^)1520 3078 y Fq(k)r(;)1581 3060 y Fp(^)1579 3078 y Fq(k)1649 3048 y Fw(=)27 b Fu(A)1827 3060 y Fp(^)1825 3078 y Fq(k)r(;)1886 3060 y Fp(^)1884 3078 y Fq(k)1948 3048 y Fr(\010)c Fu(Ae)2166 3063 y Fq(k)2209 3048 y Fu(A:)94 3247 y Fw(Th)m(us,)42 b(the)e(algebra)d Fu(R)q Fr(hh)1073 3222 y Fm(e)1051 3247 y Fu(A)1126 3259 y Fp(^)1124 3276 y Fq(k)q(;)1184 3259 y Fp(^)1182 3276 y Fq(k)1225 3247 y Fr(ii)h Fw(is)g(generated)i(b)m(y)g(the)f(arro)m(ws) g Fu(c)f Fr(2)h Fu(Q)2792 3262 y Fp(1)2858 3247 y Fr(\\)27 b Fu(A)3026 3259 y Fp(^)3024 3276 y Fq(k)q(;)3084 3259 y Fp(^)3082 3276 y Fq(k)3163 3247 y Fw(and)39 b([)p Fu(ba)p Fw(])h(for)94 3367 y Fu(a)28 b Fr(2)g Fu(Q)344 3382 y Fp(1)397 3367 y Fr(\\)12 b Fu(e)520 3382 y Fq(k)563 3367 y Fu(A)28 b Fw(and)g Fu(b)g Fr(2)g Fu(Q)1089 3382 y Fp(1)1141 3367 y Fr(\\)12 b Fu(Ae)1337 3382 y Fq(k)1381 3367 y Fw(.)42 b(The)29 b(follo)m(wing)c(fact)j(is)f(immediate)e(from)i(the)h (de\014nitions.)94 3386 y SDict begin H.S end 94 3386 a 94 3386 a SDict begin 14 H.A end 94 3386 a 94 3386 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.6.2) cvn H.B /DEST pdfmark end 94 3386 a 161 x FA(Lemma)50 b(6.2.)e Fv(The)c(c)-5 b(orr)g(esp)g(ondenc)g(e)43 b(sending)h(e)-5 b(ach)45 b Fu(c)h Fr(2)h Fu(Q)2487 3562 y Fp(1)2556 3547 y Fr(\\)30 b Fu(A)2727 3559 y Fp(^)2725 3576 y Fq(k)r(;)2786 3559 y Fp(^)2784 3576 y Fq(k)2871 3547 y Fv(to)45 b(itself,)j(and)c(e)-5 b(ach)94 3672 y(gener)g(ator)35 b Fw([)p Fu(ba)p Fw(])h Fv(to)f Fu(ba)g Fv(extends)f(to)h(an)g(isomorphism)e(of)i Fu(R)q Fv(-algebr)-5 b(as)1383 3853 y Fu(R)q Fr(hh)1558 3828 y Fm(e)1536 3853 y Fu(A)1611 3865 y Fp(^)1609 3882 y Fq(k)q(;)1669 3865 y Fp(^)1667 3882 y Fq(k)1709 3853 y Fr(ii)28 b(!)f Fu(R)q Fr(hh)p Fu(A)p Fr(ii)2247 3865 y Fp(^)2245 3882 y Fq(k)r(;)2306 3865 y Fp(^)2304 3882 y Fq(k)2346 3853 y Fu(:)194 4065 y Fw(Let)43 b Fu(u)i Fr(7!)g Fw([)p Fu(u)p Fw(])e(denote)h(the)f(isomorphism)e Fu(R)q Fr(hh)p Fu(A)p Fr(ii)2167 4077 y Fp(^)2165 4094 y Fq(k)r(;)2226 4076 y Fp(^)2224 4094 y Fq(k)2311 4065 y Fr(!)k Fu(R)q Fr(hh)2631 4039 y Fm(e)2609 4065 y Fu(A)2684 4077 y Fp(^)2682 4094 y Fq(k)r(;)2743 4076 y Fp(^)2741 4094 y Fq(k)2783 4065 y Fr(ii)d Fw(in)m(v)m(erse)j(of)d(that)h(in)94 4190 y(Lemma)445 4190 y SDict begin H.S end 445 4190 a Fw(6.2)570 4126 y SDict begin H.R end 570 4126 a 570 4190 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.6.2) cvn H.B /ANN pdfmark end 570 4190 a Fw(.)52 b(It)35 b(acts)h(in)e(the)i (same)f(w)m(a)m(y)i(as)e(the)h(corresp)s(ondence)i Fu(S)g Fr(7!)32 b Fw([)p Fu(S)6 b Fw(])35 b(in)f(\()3212 4190 y SDict begin H.S end 3212 4190 a Fw(5.8)3337 4126 y SDict begin H.R end 3337 4126 a 3337 4190 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.8) cvn H.B /ANN pdfmark end 3337 4190 a Fw(\):)49 b([)p Fu(u)p Fw(])35 b(is)94 4306 y(obtained)e(b)m(y)h(substituting)f([)p Fu(a)1254 4321 y Fq(p)1294 4306 y Fu(a)1345 4321 y Fq(p)p Fp(+1)1475 4306 y Fw(])g(for)g(eac)m(h)h(factor)f Fu(a)2236 4321 y Fq(p)2276 4306 y Fu(a)2327 4321 y Fq(p)p Fp(+1)2490 4306 y Fw(with)g Fu(t)p Fw(\()p Fu(a)2837 4321 y Fq(p)2877 4306 y Fw(\))c(=)g Fu(h)p Fw(\()p Fu(a)3194 4321 y Fq(p)p Fp(+1)3324 4306 y Fw(\))g(=)g Fu(k)36 b Fw(of)94 4422 y(an)m(y)e(path)e Fu(a)557 4437 y Fp(1)614 4422 y Fr(\001)17 b(\001)g(\001)d Fu(a)797 4437 y Fq(d)871 4422 y Fw(o)s(ccurring)31 b(in)h(the)h(path)g(expansion)g(of)f Fu(u)p Fw(.)94 4442 y SDict begin H.S end 94 4442 a 94 4442 a SDict begin 14 H.A end 94 4442 a 94 4442 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.6.3) cvn H.B /DEST pdfmark end 94 4442 a 160 x FA(Lemma)48 b(6.3.)f Fv(The)42 b(c)-5 b(orr)g(esp)g(ondenc)g(e)42 b Fu(u)h Fr(7!)f Fw([)p Fu(u)p Fw(])h Fv(induc)-5 b(es)43 b(an)g(epimorphism)e(of)i Fu(R)q Fv(-algebr)-5 b(as)94 4733 y Fr(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\))432 4745 y Fp(^)430 4762 y Fq(k)r(;)491 4745 y Fp(^)489 4762 y Fq(k)559 4733 y Fr(!)27 b(P)8 b Fw(\()824 4708 y Fm(e)801 4733 y Fu(A)q(;)932 4708 y Fm(e)919 4733 y Fu(S)d Fw(\))1024 4745 y Fp(^)1022 4762 y Fq(k)r(;)1083 4745 y Fp(^)1081 4762 y Fq(k)1123 4733 y Fv(.)94 4916 y(Pr)-5 b(o)g(of.)42 b Fw(It)32 b(is)g(enough)h(to)g(pro)m(v)m(e)g(the)g(follo)m(wing)d(t)m (w)m(o)k(facts:)94 5092 y(\(6.3\))1182 5092 y SDict begin H.S end 1182 5092 a 1182 5092 a SDict begin 14 H.A end 1182 5092 a 1182 5092 a SDict begin [ /View [/XYZ H.V] /Dest (equation.6.3) cvn H.B /DEST pdfmark end 1182 5092 a Fu(R)q Fr(hh)1357 5067 y Fm(e)1335 5092 y Fu(A)p Fr(ii)1488 5104 y Fp(^)1486 5121 y Fq(k)q(;)1546 5104 y Fp(^)1544 5121 y Fq(k)1614 5092 y Fw(=)27 b Fu(R)q Fr(hh)1892 5067 y Fm(e)1870 5092 y Fu(A)1945 5104 y Fp(^)1943 5121 y Fq(k)q(;)2003 5104 y Fp(^)2001 5121 y Fq(k)2044 5092 y Fr(ii)21 b Fw(+)h Fu(J)9 b Fw(\()2355 5067 y Fm(e)2342 5092 y Fu(S)d Fw(\))2448 5104 y Fp(^)2446 5121 y Fq(k)r(;)2507 5104 y Fp(^)2505 5121 y Fq(k)2547 5092 y Fw(;)94 5341 y(\(6.4\))1208 5341 y SDict begin H.S end 1208 5341 a 1208 5341 a SDict begin 14 H.A end 1208 5341 a 1208 5341 a SDict begin [ /View [/XYZ H.V] /Dest (equation.6.4) cvn H.B /DEST pdfmark end 1208 5341 a Fw([)p Fu(J)j Fw(\()p Fu(S)d Fw(\))1442 5353 y Fp(^)1440 5370 y Fq(k)q(;)1500 5353 y Fp(^)1498 5370 y Fq(k)1541 5341 y Fw(])27 b Fr(\022)i Fu(R)q Fr(hh)1876 5316 y Fm(e)1854 5341 y Fu(A)1929 5353 y Fp(^)1927 5370 y Fq(k)q(;)1987 5353 y Fp(^)1985 5370 y Fq(k)2027 5341 y Fr(ii)22 b(\\)g Fu(J)9 b Fw(\()2329 5316 y Fm(e)2316 5341 y Fu(S)d Fw(\))2422 5353 y Fp(^)2420 5370 y Fq(k)r(;)2481 5353 y Fp(^)2479 5370 y Fq(k)2521 5341 y Fu(:)194 5539 y Fw(T)-8 b(o)43 b(sho)m(w)h(\()638 5539 y SDict begin H.S end 638 5539 a Fw(6.3)762 5475 y SDict begin H.R end 762 5475 a 762 5539 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.6.3) cvn H.B /ANN pdfmark end 762 5539 a Fw(\),)i(w)m(e)e(note)f(that)f(if)g(a)g(path)f Fm(e)-53 b Fu(a)1955 5554 y Fp(1)2011 5539 y Fr(\001)17 b(\001)g(\001)c Fm(e)-53 b Fu(a)2195 5554 y Fq(d)2281 5539 y Fr(2)45 b Fu(R)q Fr(hh)2567 5514 y Fm(e)2545 5539 y Fu(A)p Fr(ii)2698 5551 y Fp(^)2696 5569 y Fq(k)q(;)2756 5551 y Fp(^)2754 5569 y Fq(k)2839 5539 y Fw(do)s(es)e(not)g(b)s(elong)f (to)94 5687 y Fu(R)q Fr(hh)269 5662 y Fm(e)247 5687 y Fu(A)322 5699 y Fp(^)320 5716 y Fq(k)r(;)381 5699 y Fp(^)379 5716 y Fq(k)421 5687 y Fr(ii)f Fw(then)h(it)f(m)m(ust)h(con)m(tain)f (one)h(or)f(more)g(factors)h(of)f(the)h(form)f Fu(a)2966 5651 y Fq(?)3005 5687 y Fu(b)3046 5651 y Fq(?)3128 5687 y Fw(with)g Fu(h)p Fw(\()p Fu(a)p Fw(\))j(=)p eop end end %%Page: 27 27 TeXDict begin HPSdict begin 27 26 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.27) cvn H.B /DEST pdfmark end 94 218 a 1179 w Fs(QUIVERS)31 b(WITH)i(POTENTIALS)g(I)1102 b(27)94 417 y Fu(t)p Fw(\()p Fu(b)p Fw(\))29 b(=)e Fu(k)s Fw(.)44 b(In)33 b(view)g(of)f(\()999 417 y SDict begin H.S end 999 417 a Fw(5.8)1123 353 y SDict begin H.R end 1123 353 a 1123 417 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.8) cvn H.B /ANN pdfmark end 1123 417 a Fw(\))h(and)f(\()1421 417 y SDict begin H.S end 1421 417 a Fw(5.9)1546 353 y SDict begin H.R end 1546 353 a 1546 417 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.9) cvn H.B /ANN pdfmark end 1546 417 a Fw(\),)h(w)m(e)g(ha)m(v)m(e)94 610 y(\(6.5\))1397 610 y SDict begin H.S end 1397 610 a 1397 610 a SDict begin 14 H.A end 1397 610 a 1397 610 a SDict begin [ /View [/XYZ H.V] /Dest (equation.6.5) cvn H.B /DEST pdfmark end 1397 610 a Fu(a)1448 569 y Fq(?)1487 610 y Fu(b)1528 569 y Fq(?)1596 610 y Fw(=)27 b Fu(@)1750 626 y Fp([)p Fq(ba)p Fp(])1875 585 y Fm(e)1862 610 y Fu(S)h Fr(\000)23 b Fu(@)2101 626 y Fp([)p Fq(ba)p Fp(])2212 610 y Fw([)p Fu(S)6 b Fw(])p Fu(:)94 810 y Fw(Substituting)32 b(this)h(expression)h(for)e (eac)m(h)h(factor)f Fu(a)2012 774 y Fq(?)2052 810 y Fu(b)2093 774 y Fq(?)2133 810 y Fw(,)h(w)m(e)g(see)h(that)d Fm(e)-53 b Fu(a)2757 825 y Fp(1)2813 810 y Fr(\001)17 b(\001)g(\001)c Fm(e)-53 b Fu(a)2997 825 y Fq(d)3065 810 y Fr(2)28 b Fu(R)q Fr(hh)3334 785 y Fm(e)3312 810 y Fu(A)3387 822 y Fp(^)3385 840 y Fq(k)r(;)3446 822 y Fp(^)3444 840 y Fq(k)3486 810 y Fr(ii)22 b Fw(+)94 958 y Fu(J)9 b Fw(\()208 933 y Fm(e)195 958 y Fu(S)d Fw(\))301 970 y Fp(^)299 987 y Fq(k)r(;)360 970 y Fp(^)358 987 y Fq(k)400 958 y Fw(,)33 b(as)f(desired.)194 1083 y(T)-8 b(o)39 b(sho)m(w)i(\()631 1083 y SDict begin H.S end 631 1083 a Fw(6.4)755 1019 y SDict begin H.R end 755 1019 a 755 1083 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.6.4) cvn H.B /ANN pdfmark end 755 1083 a Fw(\),)g(w)m(e)f(note)g(that)f Fu(J)9 b Fw(\()p Fu(S)d Fw(\))1660 1095 y Fp(^)1658 1113 y Fq(k)q(;)1718 1095 y Fp(^)1716 1113 y Fq(k)1798 1083 y Fw(is)38 b(easily)h(seen)h(to) f(b)s(e)h(the)f(closure)h(of)e(the)i(ideal)94 1209 y(in)34 b Fu(R)q Fr(hh)p Fu(A)p Fr(ii)514 1221 y Fp(^)513 1238 y Fq(k)q(;)573 1220 y Fp(^)571 1238 y Fq(k)647 1209 y Fw(generated)h(b)m(y)g(the)f(elemen)m(ts)g Fu(@)1850 1224 y Fq(c)1885 1209 y Fu(S)40 b Fw(for)33 b(all)f(arro)m(ws)j Fu(c)29 b Fr(2)i Fu(Q)2831 1224 y Fp(1)2904 1209 y Fw(with)j Fu(t)p Fw(\()p Fu(c)p Fw(\))c Fr(6)p Fw(=)f Fu(k)37 b Fw(and)94 1334 y Fu(h)p Fw(\()p Fu(c)p Fw(\))e Fr(6)p Fw(=)g Fu(k)s Fw(,)i(together)g(with)g(the)g(elemen)m(ts)g(\()p Fu(@)1812 1349 y Fq(a)1854 1334 y Fu(S)6 b Fw(\))p Fu(a)2009 1298 y Fn(0)2069 1334 y Fw(for)36 b Fu(a;)17 b(a)2368 1298 y Fn(0)2426 1334 y Fr(2)35 b Fu(Q)2604 1349 y Fp(1)2669 1334 y Fr(\\)25 b Fu(e)2805 1349 y Fq(k)2848 1334 y Fu(A)36 b Fw(,)i(and)f Fu(b)3257 1298 y Fn(0)3281 1334 y Fw(\()p Fu(@)3370 1349 y Fq(b)3405 1334 y Fu(S)6 b Fw(\))36 b(for)94 1450 y Fu(b;)17 b(b)220 1414 y Fn(0)272 1450 y Fr(2)28 b Fu(Q)443 1465 y Fp(1)505 1450 y Fr(\\)23 b Fu(Ae)712 1465 y Fq(k)755 1450 y Fw(.)43 b(Let)33 b(us)g(apply)f(the)h(map)f Fu(u)27 b Fr(7!)h Fw([)p Fu(u)p Fw(])k(to)g(these)i(generators.)44 b(First,)32 b(w)m(e)h(ha)m(v)m(e:)94 1643 y(\(6.6\))1620 1643 y SDict begin H.S end 1620 1643 a 1620 1643 a SDict begin 14 H.A end 1620 1643 a 1620 1643 a SDict begin [ /View [/XYZ H.V] /Dest (equation.6.6) cvn H.B /DEST pdfmark end 1620 1643 a Fw([)p Fu(@)1698 1658 y Fq(c)1733 1643 y Fu(S)6 b Fw(])28 b(=)f Fu(@)2008 1658 y Fq(c)2057 1618 y Fm(e)2043 1643 y Fu(S)6 b(:)94 1822 y Fw(With)32 b(a)h(little)d(bit)i (more)f(w)m(ork)j(\(using)e(\()1626 1822 y SDict begin H.S end 1626 1822 a Fw(6.5)1751 1758 y SDict begin H.R end 1751 1758 a 1751 1822 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.6.5) cvn H.B /ANN pdfmark end 1751 1822 a Fw(\)\),)g(w)m(e)i(obtain)1063 2023 y([\()p Fu(@)1179 2038 y Fq(a)1221 2023 y Fu(S)6 b Fw(\))p Fu(a)1376 1982 y Fn(0)1399 2023 y Fw(])28 b(=)1588 1928 y Fm(X)1558 2144 y Fq(t)p Fp(\()p Fq(b)p Fp(\)=)p Fq(k)1762 2023 y Fw(\()p Fu(@)1851 2038 y Fp([)p Fq(ba)p Fp(])1962 2023 y Fw([)p Fu(S)6 b Fw(]\)[)p Fu(ba)2239 1982 y Fn(0)2263 2023 y Fw(])1454 2313 y(=)1588 2218 y Fm(X)1558 2434 y Fq(t)p Fp(\()p Fq(b)p Fp(\)=)p Fq(k)1762 2313 y Fw(\()p Fu(@)1851 2328 y Fp([)p Fq(ba)p Fp(])1976 2287 y Fm(e)1962 2313 y Fu(S)28 b Fr(\000)23 b Fu(a)2201 2272 y Fq(?)2240 2313 y Fu(b)2281 2272 y Fq(?)2321 2313 y Fw(\)[)p Fu(ba)2478 2272 y Fn(0)2502 2313 y Fw(])94 2313 y SDict begin H.S end 94 2313 a 94 2313 a SDict begin 17.5 H.A end 94 2313 a 94 2313 a SDict begin [ /View [/XYZ H.V] /Dest (equation.6.7) cvn H.B /DEST pdfmark end 94 2313 a Fw(\(6.7\))1454 2602 y(=)1588 2508 y Fm(X)1558 2724 y Fq(t)p Fp(\()p Fq(b)p Fp(\)=)p Fq(k)1762 2602 y Fw(\()p Fu(@)1851 2618 y Fp([)p Fq(ba)p Fp(])1976 2577 y Fm(e)1962 2602 y Fu(S)6 b Fw(\)[)p Fu(ba)2185 2561 y Fn(0)2209 2602 y Fw(])22 b Fr(\000)h Fu(a)2409 2561 y Fq(?)2448 2602 y Fu(@)2499 2617 y Fq(a)2536 2598 y Ff(0)2560 2591 y Fg(?)2613 2577 y Fm(e)2600 2602 y Fu(S)6 b(;)94 2890 y Fw(and)1031 3069 y([)p Fu(b)1099 3028 y Fn(0)1123 3069 y Fw(\()p Fu(@)1212 3084 y Fq(b)1247 3069 y Fu(S)g Fw(\)])27 b(=)1550 2975 y Fm(X)1509 3191 y Fq(h)p Fp(\()p Fq(a)p Fp(\)=)p Fq(k)1735 3069 y Fw([)p Fu(b)1803 3028 y Fn(0)1827 3069 y Fu(a)p Fw(]\()p Fu(@)1994 3085 y Fp([)p Fq(ba)p Fp(])2106 3069 y Fw([)p Fu(S)6 b Fw(]\))1405 3359 y(=)1550 3265 y Fm(X)1509 3480 y Fq(h)p Fp(\()p Fq(a)p Fp(\)=)p Fq(k)1735 3359 y Fw([)p Fu(b)1803 3318 y Fn(0)1827 3359 y Fu(a)p Fw(]\()p Fu(@)1994 3375 y Fp([)p Fq(ba)p Fp(])2119 3334 y Fm(e)2106 3359 y Fu(S)28 b Fr(\000)22 b Fu(a)2344 3318 y Fq(?)2384 3359 y Fu(b)2425 3318 y Fq(?)2465 3359 y Fw(\))94 3359 y SDict begin H.S end 94 3359 a 94 3359 a SDict begin 17.5 H.A end 94 3359 a 94 3359 a SDict begin [ /View [/XYZ H.V] /Dest (equation.6.8) cvn H.B /DEST pdfmark end 94 3359 a Fw(\(6.8\))1405 3649 y(=)1550 3554 y Fm(X)1509 3770 y Fq(h)p Fp(\()p Fq(a)p Fp(\)=)p Fq(k)1735 3649 y Fw([)p Fu(b)1803 3608 y Fn(0)1827 3649 y Fu(a)p Fw(]\()p Fu(@)1994 3664 y Fp([)p Fq(ba)p Fp(])2119 3624 y Fm(e)2106 3649 y Fu(S)5 b Fw(\))23 b Fr(\000)f Fw(\()p Fu(@)2420 3664 y Fq(b)2450 3645 y Ff(0)2473 3638 y Fg(?)2527 3624 y Fm(e)2513 3649 y Fu(S)6 b Fw(\))p Fu(b)2658 3608 y Fq(?)2698 3649 y Fu(:)94 3943 y Fw(This)33 b(implies)d(the)j(desired)g(inclusion)e(in)h(\()1706 3943 y SDict begin H.S end 1706 3943 a Fw(6.4)1831 3878 y SDict begin H.R end 1831 3878 a 1831 3943 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.6.4) cvn H.B /ANN pdfmark end 1831 3943 a Fw(\).)1688 b Fh(\003)194 4136 y Fw(T)-8 b(o)33 b(\014nish)f(the)h(pro)s(of)f(of)g(Prop)s(osition)1657 4136 y SDict begin H.S end 1657 4136 a Fw(6.1)1782 4071 y SDict begin H.R end 1782 4071 a 1782 4136 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.6.1) cvn H.B /ANN pdfmark end 1782 4136 a Fw(,)g(it)g(is)g(enough)h(to)f(sho)m(w)i(that)e(the) i(epimorphism)94 4252 y(in)43 b(Lemma)578 4252 y SDict begin H.S end 578 4252 a Fw(6.3)703 4188 y SDict begin H.R end 703 4188 a 703 4252 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.6.3) cvn H.B /ANN pdfmark end 703 4252 a 43 w Fw(\(let)g(us)h(denote)g(it)e(b)m(y)i Fu(\013)q Fw(\))f(is)g(in)g(fact)g(an)g(isomorphism.)74 b(T)-8 b(o)43 b(do)g(this,)j(w)m(e)94 4382 y(construct)d(the)e(left)f(in)m(v)m (erse)j(algebra)d(homomorphism)e Fu(\014)48 b Fw(:)42 b Fr(P)8 b Fw(\()2576 4357 y Fm(e)2553 4382 y Fu(A;)2683 4357 y Fm(e)2670 4382 y Fu(S)e Fw(\))2776 4394 y Fp(^)2774 4412 y Fq(k)q(;)2834 4394 y Fp(^)2832 4412 y Fq(k)2917 4382 y Fr(!)41 b(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\))3396 4394 y Fp(^)3394 4412 y Fq(k)r(;)3455 4394 y Fp(^)3453 4412 y Fq(k)3536 4382 y Fw(\(so)94 4507 y(that)31 b Fu(\014)6 b(\013)31 b Fw(is)f(the)h(iden)m(tit)m(y)f(map)g(on)h Fr(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\))1765 4519 y Fp(^)1763 4537 y Fq(k)q(;)1823 4519 y Fp(^)1821 4537 y Fq(k)1864 4507 y Fw(\).)42 b(W)-8 b(e)31 b(de\014ne)h Fu(\014)k Fw(as)31 b(the)g(comp)s(osition)d(of)j(three)94 4684 y(maps.)42 b(First,)28 b(w)m(e)h(apply)f(the)g(epimorphism)e Fr(P)8 b Fw(\()1927 4659 y Fm(e)1904 4684 y Fu(A)q(;)2035 4659 y Fm(e)2022 4684 y Fu(S)d Fw(\))2127 4696 y Fp(^)2125 4714 y Fq(k)r(;)2186 4696 y Fp(^)2184 4714 y Fq(k)2254 4684 y Fr(!)27 b(P)8 b Fw(\()2505 4630 y Fm(e)2519 4659 y(e)2496 4684 y Fu(A;)2618 4630 y Fm(e)2626 4659 y(e)2613 4684 y Fu(S)e Fw(\))2719 4696 y Fp(^)2717 4714 y Fq(k)q(;)2777 4696 y Fp(^)2775 4714 y Fq(k)2846 4684 y Fw(de\014ned)29 b(in)e(the)i(same)94 4804 y(w)m(a)m(y)g(as)f Fu(\013)q Fw(.)41 b(Remem)m(b)s(ering)26 b(the)i(pro)s(of)e(of)h(Theorem)2072 4804 y SDict begin H.S end 2072 4804 a Fw(5.7)2197 4740 y SDict begin H.R end 2197 4740 a 2197 4804 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.5.7) cvn H.B /ANN pdfmark end 2197 4804 a 27 w Fw(and)g(using)g(the)h(notation)e(in)m(tro)s (duced)94 4958 y(there,)47 b(w)m(e)d(then)g(apply)f(the)h(isomorphism)d Fr(P)8 b Fw(\()1934 4904 y Fm(e)1948 4933 y(e)1925 4958 y Fu(A;)2047 4904 y Fm(e)2055 4933 y(e)2042 4958 y Fu(S)e Fw(\))2148 4970 y Fp(^)2146 4987 y Fq(k)q(;)2206 4970 y Fp(^)2204 4987 y Fq(k)2292 4958 y Fr(!)46 b(P)8 b Fw(\()p Fu(A)30 b Fr(\010)g Fu(C)r(;)17 b(S)34 b Fw(+)29 b Fu(T)14 b Fw(\))3189 4970 y Fp(^)3187 4987 y Fq(k)q(;)3247 4970 y Fp(^)3245 4987 y Fq(k)3331 4958 y Fw(induced)94 5083 y(b)m(y)43 b(the)f(automorphism)e Fu(')1135 5098 y Fp(3)1174 5083 y Fu(')1238 5098 y Fp(2)1277 5083 y Fu(')1341 5098 y Fp(1)1422 5083 y Fw(of)h Fu(R)q Fr(hh)p Fu(A)28 b Fr(\010)h Fu(C)7 b Fr(ii)p Fw(.)70 b(Finally)-8 b(,)42 b(w)m(e)g(apply)g(the)g (isomorphism)94 5199 y Fr(P)8 b Fw(\()p Fu(A)23 b Fr(\010)g Fu(C)r(;)17 b(S)27 b Fw(+)22 b Fu(T)14 b Fw(\))817 5211 y Fp(^)815 5229 y Fq(k)q(;)875 5211 y Fp(^)873 5229 y Fq(k)943 5199 y Fr(!)27 b(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\))1408 5211 y Fp(^)1406 5229 y Fq(k)r(;)1467 5211 y Fp(^)1465 5229 y Fq(k)1540 5199 y Fw(giv)m(en)32 b(in)g(Prop)s (osition)2433 5199 y SDict begin H.S end 2433 5199 a Fw(4.5)2558 5135 y SDict begin H.R end 2558 5135 a 2558 5199 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.5) cvn H.B /ANN pdfmark end 2558 5199 a Fw(.)194 5319 y(Since)44 b(all)d(the)j(maps)f(in)m(v)m(olv)m(ed)h(are)f(algebra)f (homomorphisms,)i(it)e(is)h(enough)h(to)f(c)m(hec)m(k)94 5435 y(that)35 b Fu(\014)6 b(\013)35 b Fw(\014xes)i(the)e(generators)g Fu(p)p Fw(\()p Fu(c)p Fw(\))g(and)g Fu(p)p Fw(\()p Fu(ba)p Fw(\))g(of)g Fr(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\))2431 5447 y Fp(^)2429 5465 y Fq(k)q(;)2489 5447 y Fp(^)2487 5465 y Fq(k)2529 5435 y Fw(,)36 b(where)g Fu(p)f Fw(is)f(the)i(pro)5 b(jection)94 5561 y Fu(R)q Fr(hh)p Fu(A)p Fr(ii)42 b(!)g(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\),)43 b(and)e Fu(a;)17 b(b;)g(c)41 b Fw(ha)m(v)m(e)i(the)e(same)g(meaning)f(as)h(ab)s(o)m(v)m (e.)70 b(This)41 b(is)g(done)g(b)m(y)94 5677 y(direct)33 b(tracing)e(of)i(the)g(de\014nitions.)2136 b Fh(\003)94 5773 y SDict begin H.S end 94 5773 a 94 5773 a SDict begin 14 H.A end 94 5773 a 94 5773 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.6.4) cvn H.B /DEST pdfmark end 94 5773 a eop end end %%Page: 28 28 TeXDict begin HPSdict begin 28 27 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.28) cvn H.B /DEST pdfmark end 94 218 a Fs(28)492 b(HARM)25 b(DERKSEN,)f(JERZY)i(WEYMAN,)f(AND)f(ANDREI)g(ZELEVINSKY)94 417 y FA(Prop)s(osition)h(6.4.)35 b Fv(In)25 b(the)h(situation)h(of)e (Pr)-5 b(op)g(osition)2182 417 y SDict begin H.S end 2182 417 a Fv(6.1)2311 352 y SDict begin H.R end 2311 352 a 2311 417 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.6.1) cvn H.B /ANN pdfmark end 2311 417 a Fv(,)28 b(if)e(the)g(Jac)-5 b(obian)25 b(algebr)-5 b(a)25 b Fr(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\))94 547 y Fv(is)35 b(\014nite-dimensional)e(then)h(so)h (is)f Fr(P)8 b Fw(\()1577 522 y Fm(e)1554 547 y Fu(A)q(;)1685 522 y Fm(e)1672 547 y Fu(S)e Fw(\))p Fv(.)94 723 y(Pr)-5 b(o)g(of.)42 b Fw(W)-8 b(e)48 b(start)g(b)m(y)h(sho)m(wing)f(that)g (\014nite)g(dimensionalit)m(y)d(of)i Fr(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\))48 b(follo)m(ws)f(from)g(a)94 839 y(seemingly)32 b(w)m(eak)m(er)j(condition.)94 858 y SDict begin H.S end 94 858 a 94 858 a SDict begin 14 H.A end 94 858 a 94 858 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.6.5) cvn H.B /DEST pdfmark end 94 858 a 156 x FA(Lemma)46 b(6.5.)g Fv(L)-5 b(et)41 b Fu(J)50 b Fr(\022)40 b Fl(m)p Fw(\()p Fu(A)p Fw(\))i Fv(b)-5 b(e)41 b(a)h(close)-5 b(d)40 b(ide)-5 b(al)41 b(in)g Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fv(.)64 b(Then)41 b(the)g(quotient)h(alge-)94 1130 y(br)-5 b(a)43 b Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fu(=J)51 b Fv(is)43 b(\014nite)f(dimensional)f(pr)-5 b(ovide)g(d)42 b(the)h(sub)-5 b(algebr)g(a)42 b Fu(R)q Fr(hh)p Fu(A)p Fr(ii)2988 1142 y Fp(^)2986 1160 y Fq(k)q(;)3046 1142 y Fp(^)3044 1160 y Fq(k)3086 1130 y Fu(=J)3191 1142 y Fp(^)3189 1160 y Fq(k)q(;)3249 1142 y Fp(^)3247 1160 y Fq(k)3332 1130 y Fv(is)h(\014nite)94 1256 y(dimensional.)f(In)30 b(p)-5 b(articular,)32 b(the)f(Jac)-5 b(obian)30 b(algebr)-5 b(a)31 b Fr(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\))31 b Fv(is)f(\014nite-dimensional)f(if)i(and)94 1372 y(only)k(if)g(so)f(is)h (the)g(sub)-5 b(algebr)g(a)34 b Fr(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\))1592 1384 y Fp(^)1590 1401 y Fq(k)q(;)1650 1384 y Fp(^)1648 1401 y Fq(k)1691 1372 y Fv(.)94 1556 y(Pr)-5 b(o)g(of.)42 b Fw(Similarly)28 b(to)k(\()964 1556 y SDict begin H.S end 964 1556 a Fw(6.1)1089 1492 y SDict begin H.R end 1089 1492 a 1089 1556 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.6.1) cvn H.B /ANN pdfmark end 1089 1556 a Fw(\),)h(for)f(an)g Fu(R)q Fw(-bimo)s(dule)e Fu(B)5 b Fw(,)33 b(w)m(e)g(denote)770 1737 y Fu(B)844 1766 y Fq(k)r(;)905 1748 y Fp(^)903 1766 y Fq(k)972 1737 y Fw(=)28 b Fu(e)1121 1752 y Fq(k)1164 1737 y Fu(B)p 1243 1682 46 4 v 5 w(e)1288 1752 y Fq(k)1359 1737 y Fw(=)1462 1642 y Fm(M)1474 1854 y Fq(j)t Fn(6)p Fp(=)p Fq(k)1629 1737 y Fu(B)1703 1752 y Fq(k)r(;j)1798 1737 y Fu(;)114 b(B)2015 1749 y Fp(^)2013 1766 y Fq(k)r(;k)2142 1737 y Fw(=)p 2245 1682 V 27 w Fu(e)2291 1752 y Fq(k)2333 1737 y Fu(B)5 b(e)2457 1752 y Fq(k)2528 1737 y Fw(=)2631 1642 y Fm(M)2648 1854 y Fq(i)p Fn(6)p Fp(=)p Fq(k)2799 1737 y Fu(B)2873 1752 y Fq(i;k)2959 1737 y Fu(:)94 2002 y Fw(W)-8 b(e)39 b(need)g(to)e(sho)m(w)i(that)f(if)e Fu(R)q Fr(hh)p Fu(A)p Fr(ii)1489 2014 y Fp(^)1487 2032 y Fq(k)q(;)1547 2014 y Fp(^)1545 2032 y Fq(k)1587 2002 y Fu(=J)1692 2014 y Fp(^)1690 2032 y Fq(k)r(;)1751 2014 y Fp(^)1749 2032 y Fq(k)1829 2002 y Fw(is)h(\014nite)h(dimensional)d (then)j(so)g(is)g(eac)m(h)h(of)e(the)94 2128 y(spaces)24 b Fu(R)q Fr(hh)p Fu(A)p Fr(ii)687 2157 y Fq(k)r(;)748 2139 y Fp(^)746 2157 y Fq(k)787 2128 y Fu(=J)890 2157 y Fq(k)r(;)951 2139 y Fp(^)949 2157 y Fq(k)990 2128 y Fw(,)g Fu(R)q Fr(hh)p Fu(A)p Fr(ii)1347 2140 y Fp(^)1345 2157 y Fq(k)q(;k)1445 2128 y Fu(=J)1550 2140 y Fp(^)1548 2157 y Fq(k)q(;k)1670 2128 y Fw(and)e Fu(R)q Fr(hh)p Fu(A)p Fr(ii)2153 2143 y Fq(k)r(;k)2253 2128 y Fu(=J)2356 2143 y Fq(k)r(;k)2457 2128 y Fw(.)39 b(Let)22 b(us)h(treat)e Fu(R)q Fr(hh)p Fu(A)p Fr(ii)3333 2143 y Fq(k)r(;k)3433 2128 y Fu(=J)3536 2143 y Fq(k)r(;k)3637 2128 y Fw(;)94 2253 y(the)33 b(other)g(t)m(w)m(o)g(cases)h(are)f(done)g(similarly)c (\(and)k(a)f(little)e(simpler\).)194 2369 y(Let)778 2487 y Fu(Q)855 2502 y Fp(1)917 2487 y Fr(\\)22 b Fu(A)1078 2516 y Fq(k)r(;)1139 2499 y Fp(^)1137 2516 y Fq(k)1207 2487 y Fw(=)27 b Fr(f)p Fu(a)1411 2502 y Fp(1)1451 2487 y Fu(;)17 b(:)g(:)g(:)f(;)h(a)1721 2502 y Fq(s)1757 2487 y Fr(g)p Fu(;)114 b(Q)2025 2502 y Fp(1)2087 2487 y Fr(\\)23 b Fu(A)2251 2499 y Fp(^)2249 2516 y Fq(k)q(;k)2377 2487 y Fw(=)28 b Fr(f)p Fu(b)2572 2502 y Fp(1)2612 2487 y Fu(;)17 b(:)g(:)g(:)e(;)i(b)2871 2502 y Fq(t)2901 2487 y Fr(g)p Fu(:)94 2629 y Fw(W)-8 b(e)33 b(ha)m(v)m(e)1065 2747 y Fu(R)q Fr(hh)p Fu(A)p Fr(ii)1369 2762 y Fq(k)r(;k)1497 2747 y Fw(=)27 b Fu(K)7 b(e)1735 2762 y Fq(k)1801 2747 y Fr(\010)1900 2652 y Fm(M)1920 2864 y Fq(`;m)2067 2747 y Fu(a)2118 2762 y Fq(`)2152 2747 y Fu(R)q Fr(hh)p Fu(A)p Fr(ii)2458 2759 y Fp(^)2456 2776 y Fq(k;)2515 2759 y Fp(^)2513 2776 y Fq(k)2556 2747 y Fu(b)2597 2762 y Fq(m)2664 2747 y Fu(:)94 2992 y Fw(It)29 b(follo)m(ws)f(that)h(there)g(is)f(a)h (surjectiv)m(e)h(map)e Fu(\013)h Fw(:)e Fu(K)22 b Fr(\002)15 b Fw(Mat)2302 3007 y Fq(s)p Fn(\002)p Fq(t)2419 2992 y Fw(\()p Fu(R)q Fr(hh)p Fu(A)p Fr(ii)2763 3004 y Fp(^)2761 3022 y Fq(k)q(;)2821 3004 y Fp(^)2819 3022 y Fq(k)2861 2992 y Fw(\))28 b Fr(!)f Fu(R)q Fr(hh)p Fu(A)p Fr(ii)3358 3007 y Fq(k)r(;k)3458 2992 y Fu(=J)3561 3007 y Fq(k)r(;k)94 3112 y Fw(giv)m(en)33 b(b)m(y)907 3421 y Fu(\013)q Fw(\()p Fu(c;)17 b(C)7 b Fw(\))27 b(=)g Fu(p)p Fw(\()p Fu(ce)1513 3436 y Fq(k)1578 3421 y Fw(+)1676 3340 y Fm(\000)1722 3419 y Fu(a)1773 3434 y Fp(1)1895 3419 y Fu(a)1946 3434 y Fp(2)2069 3419 y Fr(\001)17 b(\001)g(\001)98 b Fu(a)2336 3434 y Fq(s)2373 3340 y Fm(\001)2435 3421 y Fu(C)2529 3161 y Fm(0)2529 3336 y(B)2529 3396 y(B)2529 3460 y(@)2616 3223 y Fu(b)2657 3238 y Fp(1)2616 3339 y Fu(b)2657 3354 y Fp(2)2643 3433 y Fw(.)2643 3467 y(.)2643 3500 y(.)2621 3616 y Fu(b)2662 3631 y Fq(t)2697 3161 y Fm(1)2697 3336 y(C)2697 3396 y(C)2697 3460 y(A)2784 3421 y Fw(\))p Fu(;)94 3756 y Fw(where)29 b(Mat)547 3771 y Fq(s)p Fn(\002)p Fq(t)664 3756 y Fw(\()p Fu(B)5 b Fw(\))27 b(stands)i(for)d(the)i(space) g(of)f Fu(s)11 b Fr(\002)g Fu(t)28 b Fw(matrices)e(with)h(en)m(tries)h (in)e Fu(B)5 b Fw(,)29 b(and)e Fu(p)g Fw(is)g(the)94 3872 y(pro)5 b(jection)40 b Fu(R)q Fr(hh)p Fu(A)p Fr(ii)f(!)h Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fu(=J)9 b Fw(.)64 b(The)41 b(k)m(ernel)f(of)f Fu(\013)i Fw(con)m(tains)e(the)i(space)f(Mat)3287 3887 y Fq(s)p Fn(\002)p Fq(t)3404 3872 y Fw(\()p Fu(J)3498 3884 y Fp(^)3496 3902 y Fq(k)q(;)3556 3884 y Fp(^)3554 3902 y Fq(k)3596 3872 y Fw(\),)94 3997 y(hence)32 b Fu(R)q Fr(hh)p Fu(A)p Fr(ii)667 4012 y Fq(k)r(;k)767 3997 y Fu(=J)870 4012 y Fq(k)r(;k)1000 3997 y Fw(is)d(isomorphic)f(to)h(a)h (quotien)m(t)g(of)f(the)h(\014nite-dimensional)d(space)j Fu(K)24 b Fr(\002)94 4114 y Fw(Mat)271 4129 y Fq(s)p Fn(\002)p Fq(t)388 4114 y Fw(\()p Fu(R)q Fr(hh)p Fu(A)p Fr(ii)732 4126 y Fp(^)730 4143 y Fq(k;)789 4126 y Fp(^)787 4143 y Fq(k)830 4114 y Fu(=J)935 4126 y Fp(^)933 4143 y Fq(k)q(;)993 4126 y Fp(^)991 4143 y Fq(k)1033 4114 y Fw(\).)43 b(Th)m(us,)35 b Fu(R)q Fr(hh)p Fu(A)p Fr(ii)1720 4129 y Fq(k)r(;k)1820 4114 y Fu(=J)1923 4129 y Fq(k)r(;k)2056 4114 y Fw(is)d(\014nite)g(dimensional,)e(as)j(desired.)172 b Fh(\003)194 4291 y Fw(T)-8 b(o)32 b(\014nish)g(the)h(pro)s(of)e(of)h (Prop)s(osition)1654 4291 y SDict begin H.S end 1654 4291 a Fw(6.4)1779 4227 y SDict begin H.R end 1779 4227 a 1779 4291 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.6.4) cvn H.B /ANN pdfmark end 1779 4291 a Fw(,)g(supp)s(ose)h(that)f Fr(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\))32 b(is)g(\014nite)g (dimensional.)94 4421 y(Then)41 b Fr(P)8 b Fw(\()494 4396 y Fm(e)471 4421 y Fu(A)q(;)602 4396 y Fm(e)589 4421 y Fu(S)d Fw(\))694 4433 y Fp(^)692 4451 y Fq(k)q(;)752 4433 y Fp(^)750 4451 y Fq(k)832 4421 y Fw(is)39 b(\014nite)g (dimensional)e(b)m(y)j(Prop)s(osition)2411 4421 y SDict begin H.S end 2411 4421 a Fw(6.1)2535 4357 y SDict begin H.R end 2535 4357 a 2535 4421 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.6.1) cvn H.B /ANN pdfmark end 2535 4421 a Fw(.)64 b(Applying)39 b(Lemma)3411 4421 y SDict begin H.S end 3411 4421 a Fw(6.5)3535 4357 y SDict begin H.R end 3535 4357 a 3535 4421 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.6.5) cvn H.B /ANN pdfmark end 3535 4421 a 40 w Fw(to)94 4569 y(the)33 b(QP)g(\()498 4544 y Fm(e)475 4569 y Fu(A;)605 4544 y Fm(e)592 4569 y Fu(S)6 b Fw(\),)32 b(w)m(e)i(conclude)f(that)f Fr(P)8 b Fw(\()1649 4544 y Fm(e)1626 4569 y Fu(A)q(;)1757 4544 y Fm(e)1744 4569 y Fu(S)d Fw(\))33 b(is)f(\014nite)g(dimensional,)e(as)j(desired.) 348 b Fh(\003)194 4737 y Fw(Remem)m(b)s(ering)33 b(\()853 4737 y SDict begin H.S end 853 4737 a Fw(5.16)1027 4673 y SDict begin H.R end 1027 4673 a 1027 4737 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.16) cvn H.B /ANN pdfmark end 1027 4737 a Fw(\))h(and)h(using)f(Prop)s(osition)2075 4737 y SDict begin H.S end 2075 4737 a Fw(4.5)2199 4673 y SDict begin H.R end 2199 4673 a 2199 4737 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.5) cvn H.B /ANN pdfmark end 2199 4737 a Fw(,)h(w)m(e)h(see)g(that)e(Prop)s(ositions)3345 4737 y SDict begin H.S end 3345 4737 a Fw(6.1)3470 4673 y SDict begin H.R end 3470 4673 a 3470 4737 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.6.1) cvn H.B /ANN pdfmark end 3470 4737 a 34 w Fw(and)94 4853 y SDict begin H.S end 94 4853 a Fw(6.4)219 4789 y SDict begin H.R end 219 4789 a 219 4853 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.6.4) cvn H.B /ANN pdfmark end 219 4853 a 33 w Fw(ha)m(v)m(e)g(the)f (follo)m(wing)c(corollary)-8 b(.)94 4873 y SDict begin H.S end 94 4873 a 94 4873 a SDict begin 14 H.A end 94 4873 a 94 4873 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.6.6) cvn H.B /DEST pdfmark end 94 4873 a 156 x FA(Corollary)46 b(6.6.)g Fv(Supp)-5 b(ose)41 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\))41 b Fv(is)g(a)g(r)-5 b(e)g(duc)g(e)g(d)41 b(QP)h(satisfying)f Fw(\()2712 5029 y SDict begin H.S end 2712 5029 a Fw(5.1)2836 4964 y SDict begin H.R end 2836 4964 a 2836 5029 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.1) cvn H.B /ANN pdfmark end 2836 5029 a Fw(\))p Fv(,)i(and)e(let)h Fw(\()p 3325 4949 74 4 v Fu(A;)p 3442 4949 66 4 v 17 w(S)6 b Fw(\))40 b(=)94 5145 y Fu(\026)153 5160 y Fq(k)196 5145 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\))47 b Fv(b)-5 b(e)48 b(a)f(r)-5 b(e)g(duc)g(e)g(d)48 b(QP)g(obtaine)-5 b(d)47 b(fr)-5 b(om)47 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\))48 b Fv(by)g(the)f(mutation)h(at)g Fu(k)s Fv(.)84 b(Then)47 b(the)94 5266 y Fu(R)q Fv(-algebr)-5 b(as)38 b Fr(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\))916 5278 y Fp(^)914 5295 y Fq(k)q(;)974 5278 y Fp(^)972 5295 y Fq(k)1053 5266 y Fv(and)38 b Fr(P)8 b Fw(\()p 1361 5186 74 4 v Fu(A)q(;)p 1479 5186 66 4 v 17 w(S)d Fw(\))1584 5278 y Fp(^)1582 5295 y Fq(k)r(;)1643 5278 y Fp(^)1641 5295 y Fq(k)1721 5266 y Fv(ar)-5 b(e)39 b(isomorphic)e(to)h(e)-5 b(ach)38 b(other,)h(and)f Fr(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\))39 b Fv(is)94 5405 y(\014nite-dimensional)33 b(if)i(and)f(only)h(if)f(so)h(is)f Fr(P)8 b Fw(\()p 1823 5325 74 4 v Fu(A)q(;)p 1941 5325 66 4 v 17 w(S)d Fw(\))p Fv(.)194 5580 y Fw(W)-8 b(e)28 b(see)h(that)f(the)g(class)g(of)f(QPs)i (with)e(\014nite)h(dimensional)d(Jacobian)i(algebras)g(is)h(in)m(v)-5 b(arian)m(t)94 5696 y(under)34 b(m)m(utations.)42 b(Let)33 b(us)g(no)m(w)g(presen)m(t)i(another)d(suc)m(h)i(class.)94 5716 y SDict begin H.S end 94 5716 a 94 5716 a SDict begin 14 H.A end 94 5716 a 94 5716 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.6.7) cvn H.B /DEST pdfmark end 94 5716 a eop end end %%Page: 29 29 TeXDict begin HPSdict begin 29 28 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.29) cvn H.B /DEST pdfmark end 94 218 a 1179 w Fs(QUIVERS)31 b(WITH)i(POTENTIALS)g(I)1102 b(29)94 417 y FA(De\014nition)35 b(6.7.)41 b Fw(F)-8 b(or)31 b(ev)m(ery)i(QP)f(\()p Fu(A;)17 b(S)6 b Fw(\),)31 b(w)m(e)i(de\014ne)f (its)f Fv(deformation)i(sp)-5 b(ac)g(e)38 b Fw(Def)7 b(\()p Fu(A;)17 b(S)6 b Fw(\))30 b(b)m(y)94 578 y(\(6.9\))1280 578 y SDict begin H.S end 1280 578 a 1280 578 a SDict begin 14 H.A end 1280 578 a 1280 578 a SDict begin [ /View [/XYZ H.V] /Dest (equation.6.9) cvn H.B /DEST pdfmark end 1280 578 a 16 w Fw(Def)7 b(\()p Fu(A;)17 b(S)6 b Fw(\))27 b(=)g(T)-8 b(r)q(\()p Fr(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\)\))p Fu(=R)94 738 y Fw(\(see)34 b(De\014nitions)782 738 y SDict begin H.S end 782 738 a Fw(3.1)907 674 y SDict begin H.R end 907 674 a 907 738 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.3.1) cvn H.B /ANN pdfmark end 907 738 a 33 w Fw(and)1129 738 y SDict begin H.S end 1129 738 a Fw(3.4)1254 674 y SDict begin H.R end 1254 674 a 1254 738 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.3.4) cvn H.B /ANN pdfmark end 1254 738 a Fw(\).)94 823 y SDict begin H.S end 94 823 a 94 823 a SDict begin 14 H.A end 94 823 a 94 823 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.6.8) cvn H.B /DEST pdfmark end 94 823 a 91 x FA(Remark)53 b(6.8.)48 b Fw(De\014nition)1224 914 y SDict begin H.S end 1224 914 a Fw(6.7)1349 850 y SDict begin H.R end 1349 850 a 1349 914 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.6.7) cvn H.B /ANN pdfmark end 1349 914 a 46 w Fw(can)e(b)s(e)g(motiv)-5 b(ated)44 b(as)i(follo)m(ws)f(\(w)m(e)h(k)m(eep)i(the)e(follo)m(wing)94 1031 y(argumen)m(ts)33 b(informal)c(although)i(with)h(some)g(w)m(ork)h (they)g(can)g(b)s(e)f(made)g(rigorous\).)42 b(Let)33 b Fu(G)28 b Fw(=)94 1147 y(Aut)q(\()p Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fw(\))e(b)s(e)i(the)g(group)g(of)f(con)m(tin)m(uous)h Fu(R)q Fw(-algebra)f(automorphisms)f(of)h Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fw(.)41 b(Using)94 1263 y(Prop)s(osition)627 1263 y SDict begin H.S end 627 1263 a Fw(2.5)752 1199 y SDict begin H.R end 752 1199 a 752 1263 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.2.5) cvn H.B /ANN pdfmark end 752 1263 a Fw(,)h(w)m(e)g(can)e(think)g(of)g Fu(G)h Fw(as)f(an)g(in\014nite)g(dimensional)e(algebraic)g(group.)67 b(In)94 1379 y(view)26 b(of)f(De\014nition)862 1379 y SDict begin H.S end 862 1379 a Fw(3.4)987 1315 y SDict begin H.R end 987 1315 a 987 1379 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.3.4) cvn H.B /ANN pdfmark end 987 1379 a Fw(,)i Fu(G)e Fw(acts)g(naturally)f(on)h(the)h(trace)f(space)h (T)-8 b(r)q(\()p Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fw(\).)40 b(Remem)m(b)s(ering)94 1496 y(De\014nition)549 1496 y SDict begin H.S end 549 1496 a Fw(4.2)674 1431 y SDict begin H.R end 674 1431 a 674 1496 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.2) cvn H.B /ANN pdfmark end 674 1496 a Fw(,)34 b(it)f(is)g(natural)f(to)i(think)f(of)g(the)h (deformation)e(space)j(of)e(a)g(QP)h(\()p Fu(A;)17 b(S)6 b Fw(\))33 b(as)h(the)94 1612 y(normal)39 b(space)i(at)f Fu(\031)t Fw(\()p Fu(S)6 b Fw(\))40 b(of)f(the)i(orbit)e Fu(G)27 b Fr(\001)g Fu(\031)t Fw(\()p Fu(S)6 b Fw(\))40 b(in)g(the)g(am)m(bien)m(t)g(space)h Fu(\031)t Fw(\()p Fl(m)p Fw(\()p Fu(A)p Fw(\))3278 1576 y Fp(2)3318 1612 y Fw(\))f(\(recall)94 1728 y(that)47 b Fu(\031)j Fw(stands)e(for)e(the) g(natural)g(pro)5 b(jection)46 b Fu(R)q Fr(hh)p Fu(A)p Fr(ii)51 b(!)g Fw(T)-8 b(r\()p Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fw(\)\).)84 b(Arguing)45 b(as)i(in)94 1844 y(Lemma)444 1844 y SDict begin H.S end 444 1844 a Fw(4.11)618 1780 y SDict begin H.R end 618 1780 a 618 1844 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.11) cvn H.B /ANN pdfmark end 618 1844 a Fw(,)35 b(w)m(e)g(conclude)g(that)f(the)h(in\014nitesimal) c(action)i(of)h(\(the)h(Lie)e(algebra)h(of)7 b(\))33 b Fu(G)h Fw(on)94 1961 y Fu(\031)t Fw(\()p Fl(m)p Fw(\()p Fu(A)p Fw(\))416 1924 y Fp(2)456 1961 y Fw(\))f(is)f(b)m(y)h(the)g (transformations)1409 2218 y Fu(\031)t Fw(\()p Fu(u)p Fw(\))27 b Fr(7!)g Fu(\031)t Fw(\()1891 2093 y Fq(N)1851 2123 y Fm(X)1859 2335 y Fq(k)r Fp(=1)2011 2218 y Fu(b)2052 2233 y Fq(k)2095 2218 y Fu(@)2146 2233 y Fq(a)2183 2245 y Fg(k)2227 2218 y Fu(u)p Fw(\))p Fu(;)94 2473 y Fw(where)41 b Fu(Q)460 2488 y Fp(1)538 2473 y Fw(=)d Fr(f)p Fu(a)753 2488 y Fp(1)792 2473 y Fu(;)17 b(:)g(:)g(:)f(;)h(a)1062 2488 y Fq(N)1129 2473 y Fr(g)39 b Fw(is)f(the)h(set)h(of)e(arro)m(ws,)j (and)e Fu(b)2356 2488 y Fq(k)2438 2473 y Fr(2)f Fl(m)p Fw(\()p Fu(A)p Fw(\))2767 2488 y Fq(h)p Fp(\()p Fq(a)2872 2500 y Fg(k)2911 2488 y Fp(\))p Fq(;t)p Fp(\()p Fq(a)3047 2500 y Fg(k)3087 2488 y Fp(\))3157 2473 y Fw(\(this)h(is)f(w)m(ell)94 2589 y(de\014ned)j(in)e(view)h(of)f(Prop)s(osition)1440 2589 y SDict begin H.S end 1440 2589 a Fw(3.3)1564 2525 y SDict begin H.R end 1564 2525 a 1564 2589 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.3.3) cvn H.B /ANN pdfmark end 1564 2589 a Fw(\).)64 b(This)40 b(mak)m(es)g(is)f(natural)f(to)h (iden)m(tify)g(the)h(tangen)m(t)94 2705 y(space)34 b(at)f Fu(\031)t Fw(\()p Fu(S)6 b Fw(\))32 b(of)g Fu(G)22 b Fr(\001)g Fu(\031)t Fw(\()p Fu(S)6 b Fw(\))32 b(with)h Fu(\031)t Fw(\()p Fu(J)9 b Fw(\()p Fu(S)d Fw(\)\),)32 b(hence)i(to)f(iden)m(tify)f(the)h(corresp)s(onding)f(normal)94 2821 y(space)i(with)e Fu(\031)t Fw(\()p Fl(m)p Fw(\()p Fu(A)p Fw(\)\))p Fu(=\031)t Fw(\()p Fu(J)9 b Fw(\()p Fu(S)d Fw(\)\),)32 b(or)g(equiv)-5 b(alen)m(tly)d(,)32 b(with)g(the)h(space)g(Def)7 b(\()p Fu(S)f Fw(\))32 b(giv)m(en)g(b)m(y) i(\()3472 2821 y SDict begin H.S end 3472 2821 a Fw(6.9)3596 2757 y SDict begin H.R end 3596 2757 a 3596 2821 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.6.9) cvn H.B /ANN pdfmark end 3596 2821 a Fw(\).)94 2906 y SDict begin H.S end 94 2906 a 94 2906 a SDict begin 14 H.A end 94 2906 a 94 2906 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.6.9) cvn H.B /DEST pdfmark end 94 2906 a 106 x FA(Prop)s(osition)e (6.9.)39 b Fv(In)32 b(the)f(situation)h(of)g(Pr)-5 b(op)g(osition)2222 3012 y SDict begin H.S end 2222 3012 a Fv(6.1)2352 2947 y SDict begin H.R end 2352 2947 a 2352 3012 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.6.1) cvn H.B /ANN pdfmark end 2352 3012 a Fv(,)32 b(deformation)e(sp)-5 b(ac)g(es)31 b Fw(Def)7 b(\()3463 2987 y Fm(e)3441 3012 y Fu(A;)3571 2987 y Fm(e)3558 3012 y Fu(S)e Fw(\))94 3128 y Fv(and)35 b Fw(Def)6 b(\()p Fu(A;)17 b(S)6 b Fw(\))35 b Fv(ar)-5 b(e)34 b(isomorphic)g(to)h(e)-5 b(ach)34 b(other.)94 3305 y(Pr)-5 b(o)g(of.)42 b Fw(In)f(view)h(of)f(Prop)s(osition)1411 3305 y SDict begin H.S end 1411 3305 a Fw(3.5)1535 3240 y SDict begin H.R end 1535 3240 a 1535 3305 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.3.5) cvn H.B /ANN pdfmark end 1535 3305 a Fw(,)j(Def)6 b(\()p Fu(A;)17 b(S)6 b Fw(\))41 b(is)g(isomorphic)f(to)h(T)-8 b(r\()p Fr(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\))3274 3317 y Fp(^)3272 3334 y Fq(k)q(;)3332 3316 y Fp(^)3330 3334 y Fq(k)3373 3305 y Fw(\))p Fu(=R)3536 3317 y Fp(^)3534 3334 y Fq(k)q(;)3594 3316 y Fp(^)3592 3334 y Fq(k)3634 3305 y Fw(.)94 3424 y(Therefore,)34 b(our)f(assertion)f(is)g(immediate)e(from)i(Prop)s (osition)2474 3424 y SDict begin H.S end 2474 3424 a Fw(6.1)2599 3360 y SDict begin H.R end 2599 3360 a 2599 3424 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.6.1) cvn H.B /ANN pdfmark end 2599 3424 a Fw(.)958 b Fh(\003)94 3498 y SDict begin H.S end 94 3498 a 94 3498 a SDict begin 14 H.A end 94 3498 a 94 3498 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.6.10) cvn H.B /DEST pdfmark end 94 3498 a 103 x FA(De\014nition)27 b(6.10.)36 b Fw(W)-8 b(e)25 b(call)e(a)i(QP)f(\()p Fu(A;)17 b(S)6 b Fw(\))24 b Fv(rigid)35 b Fw(if)23 b(Def)7 b(\()p Fu(A;)17 b(S)6 b Fw(\))27 b(=)h Fr(f)p Fw(0)p Fr(g)p Fw(,)d(i.e.,)h(if)e(T)-8 b(r\()p Fr(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\)\))28 b(=)94 3717 y Fu(R)q Fw(.)194 3893 y(Com)m(bining)j(Prop)s(ositions)1253 3893 y SDict begin H.S end 1253 3893 a Fw(4.5)1378 3829 y SDict begin H.R end 1378 3829 a 1378 3893 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.5) cvn H.B /ANN pdfmark end 1378 3893 a 32 w Fw(and)1600 3893 y SDict begin H.S end 1600 3893 a Fw(6.9)1725 3829 y SDict begin H.R end 1725 3829 a 1725 3893 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.6.9) cvn H.B /ANN pdfmark end 1725 3893 a Fw(,)h(w)m(e)i(obtain)e(the)h(follo)m(wing)c(corollary)-8 b(.)94 3913 y SDict begin H.S end 94 3913 a 94 3913 a SDict begin 14 H.A end 94 3913 a 94 3913 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.6.11) cvn H.B /DEST pdfmark end 94 3913 a 157 x FA(Corollary)49 b(6.11.)d Fv(If)d(a)h(r)-5 b(e)g(duc)g(e)g(d)43 b(QP)g Fw(\()p Fu(A;)17 b(S)6 b Fw(\))43 b Fv(satis\014es)g Fw(\()2336 4070 y SDict begin H.S end 2336 4070 a Fw(5.1)2461 4006 y SDict begin H.R end 2461 4006 a 2461 4070 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.1) cvn H.B /ANN pdfmark end 2461 4070 a Fw(\))g Fv(and)g(is)h(rigid,)h(then)e(the)h(QP)94 4191 y Fw(\()p 132 4111 74 4 v Fu(A)q(;)p 250 4111 66 4 v 17 w(S)5 b Fw(\))28 b(=)f Fu(\026)543 4206 y Fq(k)586 4191 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\))34 b Fv(is)h(also)f(rigid.)194 4367 y Fw(Some)e(examples)h(of)f(rigid)f(and)h(non-rigid)e(QPs)k(will)c (b)s(e)j(giv)m(en)f(in)g(Section)3061 4367 y SDict begin H.S end 3061 4367 a Fw(8)3109 4303 y SDict begin H.R end 3109 4303 a 3109 4367 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (section.8) cvn H.B /ANN pdfmark end 3109 4367 a Fw(.)94 4478 y SDict begin H.S end 94 4478 a 94 4478 a SDict begin 14 H.A end 94 4478 a 94 4478 a SDict begin [ /View [/XYZ H.V] /Dest (section.7) cvn H.B /DEST pdfmark end 94 4478 a 1301 4575 a Fw(7.)48 b Fx(Nondegenera)-7 b(te)37 b(QPs)194 4750 y Fw(If)26 b(w)m(e)i(wish)e(to)g(b)s(e)h(able)f (to)g(apply)g(to)g(a)g(reduced)i(QP)e(\()p Fu(A;)17 b(S)6 b Fw(\))26 b(the)h(m)m(utation)e(at)h Fv(every)35 b Fw(v)m(ertex)94 4866 y(of)f Fu(Q)284 4881 y Fp(0)324 4866 y Fw(,)h(the)g Fu(R)q Fw(-bimo)s(dule)c Fu(A)k Fw(m)m(ust)f(satisfy)h(\()1786 4866 y SDict begin H.S end 1786 4866 a Fw(5.1)1910 4802 y SDict begin H.R end 1910 4802 a 1910 4866 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.1) cvn H.B /ANN pdfmark end 1910 4866 a Fw(\))f(at)g(all)f(v)m(ertices.)50 b(Th)m(us,)36 b(the)f(arro)m(w)f(span)h Fu(A)94 4982 y Fw(m)m(ust)f(b)s(e)f(2-acyclic)f(\(see)j(De\014nition)1520 4982 y SDict begin H.S end 1520 4982 a Fw(4.14)1694 4918 y SDict begin H.R end 1694 4918 a 1694 4982 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.14) cvn H.B /ANN pdfmark end 1694 4982 a Fw(\).)45 b(Suc)m(h)34 b(an)g(arro)m(w)f(span)h Fu(A)f Fw(can)h(b)s(e)f(enco)s(ded)h(b)m(y)h(a)94 5098 y(sk)m(ew-symmetric)30 b(in)m(teger)f(matrix)e Fu(B)33 b Fw(=)27 b Fu(B)5 b Fw(\()p Fu(A)p Fw(\))28 b(=)g(\()p Fu(b)2080 5113 y Fq(i;j)2160 5098 y Fw(\))h(with)f(ro)m(ws)i(and)f (columns)f(lab)s(eled)f(b)m(y)94 5215 y Fu(Q)171 5230 y Fp(0)211 5215 y Fw(,)33 b(b)m(y)g(setting)94 5375 y(\(7.1\))1347 5375 y SDict begin H.S end 1347 5375 a 1347 5375 a SDict begin 14 H.A end 1347 5375 a 1347 5375 a SDict begin [ /View [/XYZ H.V] /Dest (equation.7.1) cvn H.B /DEST pdfmark end 1347 5375 a Fu(b)1388 5390 y Fq(i;j)1496 5375 y Fw(=)27 b(dim)16 b Fu(A)1852 5390 y Fq(i;j)1954 5375 y Fr(\000)23 b Fw(dim)15 b Fu(A)2306 5390 y Fq(j;i)2382 5375 y Fu(:)94 5536 y Fw(Indeed,)35 b(the)e(dimensions)e(of)h(the)h(comp)s(onen)m(ts)g Fu(A)1995 5551 y Fq(i;j)2108 5536 y Fw(are)g(reco)m(v)m(ered)i(from)c Fu(B)38 b Fw(b)m(y)94 5696 y(\(7.2\))1507 5696 y SDict begin H.S end 1507 5696 a 1507 5696 a SDict begin 14 H.A end 1507 5696 a 1507 5696 a SDict begin [ /View [/XYZ H.V] /Dest (equation.7.2) cvn H.B /DEST pdfmark end 1507 5696 a 16 w Fw(dim)16 b Fu(A)1776 5711 y Fq(i;j)1884 5696 y Fw(=)27 b([)p Fu(b)2055 5711 y Fq(i;j)2136 5696 y Fw(])2163 5711 y Fp(+)2222 5696 y Fu(;)p eop end end %%Page: 30 30 TeXDict begin HPSdict begin 30 29 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.30) cvn H.B /DEST pdfmark end 94 218 a Fs(30)492 b(HARM)25 b(DERKSEN,)f(JERZY)i(WEYMAN,)f(AND)f(ANDREI)g(ZELEVINSKY)94 417 y Fw(where)34 b(w)m(e)g(use)f(the)g(notation)94 574 y(\(7.3\))1512 574 y SDict begin H.S end 1512 574 a 1512 574 a SDict begin 14 H.A end 1512 574 a 1512 574 a SDict begin [ /View [/XYZ H.V] /Dest (equation.7.3) cvn H.B /DEST pdfmark end 1512 574 a Fw([)p Fu(x)p Fw(])1621 589 y Fp(+)1708 574 y Fw(=)28 b(max)o(\()p Fu(x;)17 b Fw(0\))p Fu(:)94 639 y SDict begin H.S end 94 639 a 94 639 a SDict begin 14 H.A end 94 639 a 94 639 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.7.1) cvn H.B /DEST pdfmark end 94 639 a 105 x FA(Prop)s(osition)25 b(7.1.)35 b Fv(L)-5 b(et)27 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\))26 b Fv(b)-5 b(e)26 b(a)g Fw(2)p Fv(-acyclic)f(r)-5 b(e)g(duc)g(e)g(d)26 b(QP)g(satisfying)g Fw(\()2890 744 y SDict begin H.S end 2890 744 a Fw(4.13)3063 679 y SDict begin H.R end 3063 679 a 3063 744 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.4.13) cvn H.B /ANN pdfmark end 3063 744 a Fw(\))p Fv(,)i(and)e(supp)-5 b(ose)94 865 y(that)32 b(the)g(r)-5 b(e)g(duc)g(e)g(d)30 b(QP)i Fu(\026)1019 880 y Fq(k)1061 865 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\))27 b(=)h(\()p 1489 785 74 4 v Fu(A;)p 1606 785 66 4 v 17 w(S)5 b Fw(\))31 b Fv(obtaine)-5 b(d)31 b(fr)-5 b(om)31 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\))31 b Fv(by)g(the)g(mutation)h(at)f(some)94 987 y(vertex)39 b Fu(k)i Fv(\(se)-5 b(e)38 b(De\014nition)1144 987 y SDict begin H.S end 1144 987 a Fv(5.5)1274 922 y SDict begin H.R end 1274 922 a 1274 987 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.5.5) cvn H.B /ANN pdfmark end 1274 987 a Fv(\))g(is)g(also)g Fw(2)p Fv(-acyclic.)54 b(L)-5 b(et)39 b Fu(B)5 b Fw(\()p Fu(A)p Fw(\))35 b(=)f(\()p Fu(b)2742 1002 y Fq(i;j)2822 987 y Fw(\))39 b Fv(and)e Fu(B)5 b Fw(\()p 3208 907 74 4 v Fu(A)q Fw(\))34 b(=)g(\()p 3502 906 42 4 v Fu(b)3543 1002 y Fq(i;j)3623 987 y Fw(\))94 1103 y Fv(b)-5 b(e)35 b(the)g(skew-symmetric)f(inte)-5 b(ger)34 b(matric)-5 b(es)34 b(given)g(by)h Fw(\()2213 1103 y SDict begin H.S end 2213 1103 a Fw(7.1)2338 1039 y SDict begin H.R end 2338 1039 a 2338 1103 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.7.1) cvn H.B /ANN pdfmark end 2338 1103 a Fw(\))p Fv(.)45 b(Then)34 b(we)g(have)94 1357 y Fw(\(7.4\))505 1357 y SDict begin H.S end 505 1357 a 505 1357 a SDict begin 14 H.A end 505 1357 a 505 1357 a SDict begin [ /View [/XYZ H.V] /Dest (equation.7.4) cvn H.B /DEST pdfmark end 505 1357 a 505 1275 42 4 v Fu(b)546 1372 y Fq(i;j)654 1357 y Fw(=)757 1186 y Fm(\()838 1275 y Fr(\000)p Fu(b)956 1290 y Fq(i;j)2506 1275 y Fv(if)h Fu(i)28 b Fw(=)f Fu(k)38 b Fv(or)d Fu(j)f Fw(=)27 b Fu(k)s Fv(;)838 1444 y Fu(b)879 1459 y Fq(i;j)981 1444 y Fw(+)22 b([)p Fu(b)1147 1459 y Fq(i;k)1234 1444 y Fw(])1261 1459 y Fp(+)1355 1444 y Fw([)p Fu(b)1423 1459 y Fq(k)r(;j)1519 1444 y Fw(])1546 1459 y Fp(+)1627 1444 y Fr(\000)g Fw([)p Fr(\000)p Fu(b)1871 1459 y Fq(i;k)1959 1444 y Fw(])1986 1459 y Fp(+)2080 1444 y Fw([)p Fr(\000)p Fu(b)2225 1459 y Fq(k)r(;j)2321 1444 y Fw(])2348 1459 y Fp(+)2506 1444 y Fv(otherwise.)94 1623 y(Pr)-5 b(o)g(of.)42 b Fw(First)26 b(w)m(e)j(note)f(that)f(b)m(y)i(Prop)s(osition)1833 1623 y SDict begin H.S end 1833 1623 a Fw(4.4)1957 1558 y SDict begin H.R end 1957 1558 a 1957 1623 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.4) cvn H.B /ANN pdfmark end 1957 1623 a Fw(,)g(if)e(\()p Fu(C)r(;)17 b(T)d Fw(\))26 b(is)h(a)h(trivial)d(QP)i(then)i(dim)15 b Fu(C)3478 1638 y Fq(i;j)3586 1623 y Fw(=)94 1739 y(dim)h Fu(C)344 1754 y Fq(j;i)452 1739 y Fw(for)32 b(all)f Fu(i;)17 b(j)6 b Fw(.)43 b(In)33 b(view)g(of)f(\()1426 1739 y SDict begin H.S end 1426 1739 a Fw(5.16)1599 1675 y SDict begin H.R end 1599 1675 a 1599 1739 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.16) cvn H.B /ANN pdfmark end 1599 1739 a Fw(\),)h(this)f(implies)e(that)94 1913 y(\(7.5\))890 1913 y SDict begin H.S end 890 1913 a 890 1913 a SDict begin 14 H.A end 890 1913 a 890 1913 a SDict begin [ /View [/XYZ H.V] /Dest (equation.7.5) cvn H.B /DEST pdfmark end 890 1913 a 890 1832 42 4 v Fu(b)931 1928 y Fq(i;j)1039 1913 y Fw(=)d(dim)p 1322 1833 74 4 v 16 w Fu(A)1395 1928 y Fq(i;j)1497 1913 y Fr(\000)c Fw(dim)p 1776 1833 V 15 w Fu(A)1849 1928 y Fq(j;i)1953 1913 y Fw(=)k(dim)2258 1888 y Fm(e)2236 1913 y Fu(A)2309 1928 y Fq(i;j)2411 1913 y Fr(\000)c Fw(dim)2712 1888 y Fm(e)2690 1913 y Fu(A)2763 1928 y Fq(j;i)2839 1913 y Fu(;)94 2087 y Fw(where)34 b(\()437 2062 y Fm(e)414 2087 y Fu(A;)544 2062 y Fm(e)531 2087 y Fu(S)6 b Fw(\))27 b(=)32 b Fm(e)-59 b Fu(\026)825 2102 y Fq(k)867 2087 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\).)43 b(Using)33 b(\()1509 2087 y SDict begin H.S end 1509 2087 a Fw(5.3)1633 2023 y SDict begin H.R end 1633 2023 a 1633 2087 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.3) cvn H.B /ANN pdfmark end 1633 2087 a Fw(\),)g(w)m(e)g(obtain)616 2340 y(dim)817 2315 y Fm(e)795 2340 y Fu(A)868 2355 y Fq(i;j)976 2340 y Fw(=)1079 2170 y Fm(\()1160 2258 y Fw(dim)15 b Fu(A)1412 2273 y Fq(j;i)2412 2258 y Fw(if)31 b Fu(i)d Fw(=)g Fu(k)36 b Fw(or)c Fu(j)h Fw(=)28 b Fu(k)s Fw(;)1160 2428 y(dim)15 b Fu(A)1412 2443 y Fq(i;j)1514 2428 y Fw(+)22 b(dim)16 b Fu(A)1865 2443 y Fq(i;k)1968 2428 y Fw(dim)f Fu(A)2220 2443 y Fq(k)r(;j)2412 2428 y Fw(otherwise.)94 2593 y(T)-8 b(o)23 b(obtain)f(\()560 2593 y SDict begin H.S end 560 2593 a Fw(7.4)684 2529 y SDict begin H.R end 684 2529 a 684 2593 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.7.4) cvn H.B /ANN pdfmark end 684 2593 a Fw(\),)j(it)d(remains)f(to)i(substitute)g(these)h (expressions)g(in)m(to)e(\()2735 2593 y SDict begin H.S end 2735 2593 a Fw(7.5)2860 2529 y SDict begin H.R end 2860 2529 a 2860 2593 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.7.5) cvn H.B /ANN pdfmark end 2860 2593 a Fw(\))g(and)h(use)h(\()3297 2593 y SDict begin H.S end 3297 2593 a Fw(7.2)3421 2529 y SDict begin H.R end 3421 2529 a 3421 2593 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.7.2) cvn H.B /ANN pdfmark end 3421 2593 a Fw(\).)98 b Fh(\003)194 2759 y Fw(An)38 b(easy)h(calculation)d(using)i(the)g(ob)m(vious)g(iden)m(tit)m(y)g Fu(x)g Fw(=)f([)p Fu(x)p Fw(])2551 2774 y Fp(+)2636 2759 y Fr(\000)26 b Fw([)p Fr(\000)p Fu(x)p Fw(])2925 2774 y Fp(+)3023 2759 y Fw(sho)m(ws)40 b(that)e(the)94 2876 y(second)c(case)g(in)e(\()768 2876 y SDict begin H.S end 768 2876 a Fw(7.4)892 2811 y SDict begin H.R end 892 2811 a 892 2876 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.7.4) cvn H.B /ANN pdfmark end 892 2876 a Fw(\))h(can)g(b)s(e)f (rewritten)h(in)f(sev)m(eral)h(equiv)-5 b(alen)m(t)32 b(w)m(a)m(ys)j(as)e(follo)m(ws:)p 1146 2957 42 4 v 1146 3039 a Fu(b)1188 3054 y Fq(i;j)1296 3039 y Fw(=)27 b Fu(b)1440 3054 y Fq(i;j)1543 3039 y Fw(+)22 b(sgn\()p Fu(b)1861 3054 y Fq(i;k)1948 3039 y Fw(\))33 b([)p Fu(b)2087 3054 y Fq(i;k)2173 3039 y Fu(b)2214 3054 y Fq(k)r(;j)2310 3039 y Fw(])2337 3054 y Fp(+)1296 3190 y Fw(=)27 b Fu(b)1440 3205 y Fq(i;j)1543 3190 y Fw(+)22 b([)p Fr(\000)p Fu(b)1786 3205 y Fq(i;k)1873 3190 y Fw(])1900 3205 y Fp(+)1976 3190 y Fu(b)2017 3205 y Fq(k)r(;j)2134 3190 y Fw(+)g Fu(b)2273 3205 y Fq(i;k)2360 3190 y Fw([)p Fu(b)2428 3205 y Fq(k)r(;j)2523 3190 y Fw(])2550 3205 y Fp(+)1296 3408 y Fw(=)27 b Fu(b)1440 3423 y Fq(i;j)1543 3408 y Fw(+)1651 3341 y Fr(j)p Fu(b)1720 3356 y Fq(i;k)1806 3341 y Fr(j)p Fu(b)1875 3356 y Fq(k)r(;j)1992 3341 y Fw(+)22 b Fu(b)2131 3356 y Fq(i;k)2218 3341 y Fr(j)p Fu(b)2287 3356 y Fq(k)r(;j)2382 3341 y Fr(j)p 1651 3385 759 4 v 2006 3476 a Fw(2)2420 3408 y Fu(:)94 3613 y Fw(It)36 b(follo)m(ws)f(that)g(the)h(transformation)e Fu(B)k Fr(7!)p 1824 3533 80 4 v 32 w Fu(B)j Fw(giv)m(en)36 b(b)m(y)h(\()2374 3613 y SDict begin H.S end 2374 3613 a Fw(7.4)2498 3549 y SDict begin H.R end 2498 3549 a 2498 3613 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.7.4) cvn H.B /ANN pdfmark end 2498 3613 a Fw(\))f(coincides)f(with)h(the)g Fv(matrix)94 3730 y(mutation)46 b Fw(at)38 b Fu(k)k Fw(whic)m(h)d(pla)m(ys)g(a)f (crucial)f(part)h(in)g(the)h(theory)g(of)f(cluster)g(algebras)g(\(cf.)h ([)3537 3730 y SDict begin H.S end 3537 3730 a Fw(18)3634 3665 y SDict begin H.R end 3634 3665 a 3634 3730 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.ca1) cvn H.B /ANN pdfmark end 3634 3730 a Fw(,)94 3846 y(\(4.3\)],)33 b([)409 3846 y SDict begin H.S end 409 3846 a Fw(20)506 3782 y SDict begin H.R end 506 3782 a 506 3846 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.ca4) cvn H.B /ANN pdfmark end 506 3846 a Fw(,)g(\(2.2\),)f(\(2.5\)]\).)194 3962 y(W)-8 b(e)25 b(see)h(that)f(the)g(m)m(utations)f(of)g(2-acyclic)f (QPs)j(pro)m(vide)f(a)f(natural)g(framew)m(ork)h(for)f(matrix)94 4078 y(m)m(utations.)90 b(With)48 b(some)g(abuse)h(of)f(notation,)j(w)m (e)e(denote)g(b)m(y)g Fu(\026)2716 4093 y Fq(k)2759 4078 y Fw(\()p Fu(A)p Fw(\))f(the)h(2-acyclic)e Fu(R)q Fw(-)94 4195 y(bimo)s(dule)30 b(suc)m(h)i(that)f(the)h(sk)m(ew-symmetric)g (matrix)e Fu(B)5 b Fw(\()p Fu(\026)2313 4210 y Fq(k)2355 4195 y Fw(\()p Fu(A)p Fw(\)\))32 b(is)e(obtained)h(from)f Fu(B)5 b Fw(\()p Fu(A)p Fw(\))31 b(b)m(y)94 4311 y(the)i(m)m(utation)e (at)i Fu(k)s Fw(;)f(th)m(us,)i Fu(\026)1216 4326 y Fq(k)1258 4311 y Fw(\()p Fu(A)p Fw(\))f(is)f(determined)h(b)m(y)g Fu(A)g Fw(up)g(to)f(an)g(isomorphism.)194 4427 y(Note)j(that)f(the)h (matrix)e(m)m(utations)h(at)g(arbitrary)g(v)m(ertices)h(can)g(b)s(e)g (iterated)f(inde\014nitely)-8 b(,)94 4543 y(while)33 b(the)g(2-acyclicit)m(y)e(condition)h(\()1497 4543 y SDict begin H.S end 1497 4543 a Fw(4.13)1670 4479 y SDict begin H.R end 1670 4479 a 1670 4543 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.4.13) cvn H.B /ANN pdfmark end 1670 4543 a Fw(\))h(can)g(b)s(e)g(destro)m(y)m(ed)i(b)m(y)f(a)e(QP)i (m)m(utation.)42 b(W)-8 b(e)34 b(will)94 4660 y(study)g(QPs)g(for)e (whic)m(h)h(this)f(do)s(es)h(not)f(happ)s(en.)94 4679 y SDict begin H.S end 94 4679 a 94 4679 a SDict begin 14 H.A end 94 4679 a 94 4679 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.7.2) cvn H.B /DEST pdfmark end 94 4679 a 150 x FA(De\014nition)39 b(7.2.)k Fw(Let)34 b Fu(k)1064 4844 y Fp(1)1104 4829 y Fu(;)17 b(:)g(:)g(:)e(;)i(k)1373 4844 y Fq(`)1437 4829 y Fr(2)31 b Fu(Q)1611 4844 y Fp(0)1685 4829 y Fw(b)s(e)k(a)f(\014nite)h(sequence)i(of)d(v)m(ertices)i(suc)m(h) g(that)e Fu(k)3515 4844 y Fq(p)3586 4829 y Fr(6)p Fw(=)94 4945 y Fu(k)145 4960 y Fq(p)p Fp(+1)311 4945 y Fw(for)i Fu(p)e Fw(=)g(1)p Fu(;)17 b(:)g(:)g(:)e(;)i(`)24 b Fr(\000)h Fw(1.)55 b(W)-8 b(e)36 b(sa)m(y)i(that)e(a)f(QP)i(\()p Fu(A;)17 b(S)6 b Fw(\))36 b(is)f(\()p Fu(k)2529 4960 y Fq(`)2562 4945 y Fu(;)17 b Fr(\001)g(\001)g(\001)31 b Fu(;)17 b(k)2850 4960 y Fp(1)2889 4945 y Fw(\)-)p Fv(nonde)-5 b(gener)g(ate)42 b Fw(if)94 5062 y(all)35 b(the)i(QPs)h(\()p Fu(A;)17 b(S)6 b Fw(\))p Fu(;)17 b(\026)986 5077 y Fq(k)1023 5086 y Fo(1)1060 5062 y Fw(\()p Fu(A;)g(S)6 b Fw(\))p Fu(;)17 b(\026)1422 5077 y Fq(k)1459 5086 y Fo(2)1496 5062 y Fu(\026)1555 5077 y Fq(k)1592 5086 y Fo(1)1631 5062 y Fw(\()p Fu(A;)g(S)6 b Fw(\))p Fu(;)17 b(:)g(:)g(:)e(;)i(\026) 2167 5077 y Fq(k)2204 5089 y Fg(`)2254 5062 y Fr(\001)g(\001)g(\001)e Fu(\026)2446 5077 y Fq(k)2483 5086 y Fo(1)2521 5062 y Fw(\()p Fu(A;)i(S)6 b Fw(\))36 b(are)h(2-acyclic)e(\(hence)94 5178 y(w)m(ell-de\014ned\).)66 b(W)-8 b(e)40 b(sa)m(y)g(that)g(\()p Fu(A;)17 b(S)6 b Fw(\))39 b Fv(nonde)-5 b(gener)g(ate)46 b Fw(if)38 b(it)h(is)g(\()p Fu(k)2637 5193 y Fq(`)2670 5178 y Fu(;)17 b(:)g(:)g(:)f(;)h(k)2940 5193 y Fp(1)2979 5178 y Fw(\)-nondegenerate)94 5294 y(for)33 b(ev)m(ery)h(sequence)h(of) d(v)m(ertices)i(as)f(ab)s(o)m(v)m(e.)194 5464 y(T)-8 b(o)28 b(state)g(our)g(next)g(result)g(recall)e(the)j(terminology)c(in) m(tro)s(duced)j(b)s(efore)g(Prop)s(osition)3461 5464 y SDict begin H.S end 3461 5464 a Fw(4.15)3634 5400 y SDict begin H.R end 3634 5400 a 3634 5464 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.15) cvn H.B /ANN pdfmark end 3634 5464 a Fw(.)94 5580 y(In)k(particular,)f(for)g(a)g(giv)m(en)h (quiv)m(er)g(with)f(the)h(arro)m(w)g(span)g Fu(A)p Fw(,)g(the)g(QPs)h (on)e Fu(A)h Fw(are)g(iden)m(ti\014ed)94 5696 y(with)h(the)g(elemen)m (ts)g(of)f Fu(K)1085 5660 y Fn(C)t Fp(\()p Fq(A)p Fp(\))1238 5696 y Fw(.)94 5696 y SDict begin H.S end 94 5696 a 94 5696 a SDict begin 14 H.A end 94 5696 a 94 5696 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.7.3) cvn H.B /DEST pdfmark end 94 5696 a eop end end %%Page: 31 31 TeXDict begin HPSdict begin 31 30 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.31) cvn H.B /DEST pdfmark end 94 218 a 1179 w Fs(QUIVERS)31 b(WITH)i(POTENTIALS)g(I)1102 b(31)94 417 y FA(Prop)s(osition)39 b(7.3.)k Fv(Supp)-5 b(ose)37 b(that)h(the)f(b)-5 b(ase)37 b(\014eld)f Fu(K)45 b Fv(is)37 b(in\014nite,)g Fu(Q)h Fv(is)f(a)g Fw(2)p Fv(-acyclic)g(quiver)94 533 y(with)g(the)f(arr)-5 b(ow)36 b(sp)-5 b(an)36 b Fu(A)p Fv(,)g(a)g(se)-5 b(quenc)g(e)36 b(of)g(vertic)-5 b(es)36 b Fu(k)2119 548 y Fp(1)2158 533 y Fu(;)17 b(:)g(:)g(:)f(;)h(k)2428 548 y Fq(`)2497 533 y Fv(is)36 b(as)g(in)g(De\014nition)3311 553 y SDict begin H.S end 3311 553 a -20 x Fv(7.2)3440 469 y SDict begin H.R end 3440 469 a 3440 533 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.7.2) cvn H.B /ANN pdfmark end 3440 533 a Fv(,)h(and)94 649 y Fu(A)167 613 y Fn(0)227 649 y Fw(=)g Fu(\026)399 664 y Fq(k)436 676 y Fg(`)486 649 y Fr(\001)17 b(\001)g(\001)e Fu(\026)678 664 y Fq(k)715 673 y Fo(1)753 649 y Fw(\()p Fu(A)p Fw(\))p Fv(.)59 b(Then)38 b(ther)-5 b(e)40 b(exist)f(a)h(non-zer)-5 b(o)38 b(p)-5 b(olynomial)39 b(function)g Fu(F)50 b Fw(:)36 b Fu(K)3372 613 y Fn(C)t Fp(\()p Fq(A)p Fp(\))3562 649 y Fr(!)94 772 y Fu(K)55 b Fv(and)47 b(a)g(r)-5 b(e)g(gular)47 b(map)g Fu(G)k Fw(:)g Fu(U)10 b Fw(\()p Fu(F)k Fw(\))51 b Fr(!)g Fu(K)1821 736 y Fn(C)t Fp(\()p Fq(A)1942 712 y Ff(0)1965 736 y Fp(\))2044 772 y Fv(such)c(that)h(every)f(QP)g Fw(\()p Fu(A;)17 b(S)6 b Fw(\))47 b Fv(with)g Fu(S)57 b Fr(2)94 888 y Fu(U)10 b Fw(\()p Fu(F)k Fw(\))36 b Fv(is)e Fw(\()p Fu(k)552 903 y Fq(`)585 888 y Fu(;)17 b(:)g(:)g(:)f(;)h(k)855 903 y Fp(1)894 888 y Fw(\))p Fv(-nonde)-5 b(gener)g(ate,)33 b(and,)h(for)h(any)g(QP)g Fw(\()p Fu(A;)17 b(S)6 b Fw(\))34 b Fv(with)h Fu(S)f Fr(2)28 b Fu(U)10 b Fw(\()p Fu(F)k Fw(\))p Fv(,)36 b(the)f(QP)94 1004 y Fu(\026)153 1019 y Fq(k)190 1031 y Fg(`)241 1004 y Fr(\001)17 b(\001)g(\001)d Fu(\026)432 1019 y Fq(k)469 1028 y Fo(1)508 1004 y Fw(\()p Fu(A;)j(S)6 b Fw(\))34 b Fv(is)h(right-e)-5 b(quivalent)34 b(to)h Fw(\()p Fu(A)1820 968 y Fn(0)1843 1004 y Fu(;)17 b(G)p Fw(\()p Fu(S)6 b Fw(\)\))p Fv(.)94 1159 y(Pr)-5 b(o)g(of.)42 b Fw(W)-8 b(e)41 b(pro)s(ceed)g(b)m(y)h(induction)d(on)i Fu(`)p Fw(.)68 b(First)39 b(let)h(us)h(deal)g(with)f(the)h(case)g Fu(`)h Fw(=)f(1,)i(that)94 1275 y(is,)37 b(with)f(a)g(single)f(m)m (utation)g Fu(\026)1293 1290 y Fq(k)1335 1275 y Fw(.)55 b(Recall)34 b(that)i Fu(\026)1988 1290 y Fq(k)2031 1275 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\))33 b(=)h(\()p 2471 1195 74 4 v Fu(A;)p 2588 1195 66 4 v 17 w(S)6 b Fw(\))36 b(is)f(the)i (reduced)h(part)e(of)94 1406 y(the)k(QP)k Fm(e)-59 b Fu(\026)510 1421 y Fq(k)552 1406 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\))38 b(=)h(\()1025 1381 y Fm(e)1002 1406 y Fu(A;)1132 1381 y Fm(e)1119 1406 y Fu(S)6 b Fw(\))39 b(giv)m(en)g(b)m(y)h(\()1703 1406 y SDict begin H.S end 1703 1406 a Fw(5.3)1828 1342 y SDict begin H.R end 1828 1342 a 1828 1406 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.3) cvn H.B /ANN pdfmark end 1828 1406 a Fw(\))f(and)g(\()2139 1406 y SDict begin H.S end 2139 1406 a Fw(5.8)2264 1342 y SDict begin H.R end 2264 1342 a 2264 1406 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.8) cvn H.B /ANN pdfmark end 2264 1406 a Fw(\).)63 b(It)39 b(is)g(clear)g(from)f(the)h(de\014nition)94 1540 y(that)335 1515 y Fm(e)322 1540 y Fu(S)61 b Fw(=)593 1515 y Fm(e)575 1540 y Fu(G)o Fw(\()p Fu(S)6 b Fw(\))49 b(for)f(a)h(p)s(olynomial)c(map)1877 1515 y Fm(e)1858 1540 y Fu(G)56 b Fw(:)f Fu(K)2163 1504 y Fn(C)t Fp(\()p Fq(A)p Fp(\))2371 1540 y Fr(!)g Fu(K)2616 1504 y Fn(C)t Fp(\()2701 1487 y Fe(e)2684 1504 y Fq(A)q Fp(\))2769 1540 y Fw(.)93 b(No)m(w)49 b(let)f(us)i(apply)94 1670 y(Prop)s(osition)632 1670 y SDict begin H.S end 632 1670 a Fw(4.15)805 1606 y SDict begin H.R end 805 1606 a 805 1670 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.15) cvn H.B /ANN pdfmark end 805 1670 a 45 w Fw(to)44 b(the)h(quiv)m(er)h (with)e(the)h(arro)m(w)g(span)2432 1645 y Fm(e)2409 1670 y Fu(A)q Fw(.)80 b(W)-8 b(e)45 b(see)h(that)e(there)i(exists)94 1799 y(a)g(p)s(olynomial)c(function)j(of)g(the)h(form)f Fu(D)1734 1763 y Fq(d)1770 1772 y Fo(1)1804 1763 y Fq(;:::)o(;d)1939 1774 y Fg(N)1731 1824 y Fq(c)1762 1833 y Fo(1)1796 1824 y Fq(;:::)n(;c)1925 1835 y Fg(N)2046 1799 y Fw(on)h Fu(K)2285 1763 y Fn(C)t Fp(\()2370 1746 y Fe(e)2353 1763 y Fq(A)p Fp(\))2483 1799 y Fw(\(see)h(\()2730 1799 y SDict begin H.S end 2730 1799 a Fw(4.14)2903 1735 y SDict begin H.R end 2903 1735 a 2903 1799 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.4.14) cvn H.B /ANN pdfmark end 2903 1799 a Fw(\),)i(where)e(w)m(e)g(ha)m(v)m(e)94 1919 y(c)m(hanged)24 b(the)f(notation)e(for)g(the)i(arro)m(ws)g(to)f(a)m(v)m(oid)g(the)h (notation)e(con\015ict)h(with)g(Section)3365 1919 y SDict begin H.S end 3365 1919 a Fw(5)3414 1855 y SDict begin H.R end 3414 1855 a 3414 1919 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (section.5) cvn H.B /ANN pdfmark end 3414 1919 a Fw(\))g(suc)m(h)94 2050 y(that)34 b(the)h(reduced)g(part)f(\()p 1092 1970 74 4 v Fu(A;)p 1209 1970 66 4 v 17 w(S)5 b Fw(\))34 b(of)g(a)f(QP)i(\()1778 2025 y Fm(e)1756 2050 y Fu(A;)1886 2025 y Fm(e)1873 2050 y Fu(S)5 b Fw(\))34 b(is)f(2-acyclic)g(whenev)m(er)2952 2025 y Fm(e)2939 2050 y Fu(S)j Fr(2)30 b Fu(U)10 b Fw(\()p Fu(D)3329 2014 y Fq(d)3365 2023 y Fo(1)3400 2014 y Fq(;:::)o(;d)3535 2025 y Fg(N)3326 2074 y Fq(c)3357 2083 y Fo(1)3392 2074 y Fq(;:::)n(;c)3521 2085 y Fg(N)3596 2050 y Fw(\).)94 2193 y(F)-8 b(urthermore,)40 b(for)863 2168 y Fm(e)849 2193 y Fu(S)k Fr(2)39 b Fu(U)10 b Fw(\()p Fu(D)1256 2157 y Fq(d)1292 2166 y Fo(1)1327 2157 y Fq(;:::)n(;d)1461 2168 y Fg(N)1253 2217 y Fq(c)1284 2226 y Fo(1)1318 2217 y Fq(;:::)o(;c)1448 2228 y Fg(N)1523 2193 y Fw(\),)40 b(the)f(QP)g(\()p 2021 2113 74 4 v Fu(A;)p 2138 2113 66 4 v 17 w(S)6 b Fw(\))38 b(is)h(righ)m(t-equiv)-5 b(alen)m(t)37 b(to)h(\()p Fu(A)3326 2157 y Fn(0)3350 2193 y Fu(;)17 b(H)8 b Fw(\()3534 2168 y Fm(e)3521 2193 y Fu(S)d Fw(\)\))94 2326 y(for)34 b(some)f(regular)f(map)h Fu(H)k Fw(:)30 b Fu(U)10 b Fw(\()p Fu(D)1412 2290 y Fq(d)1448 2299 y Fo(1)1483 2290 y Fq(;:::)n(;d)1617 2301 y Fg(N)1409 2351 y Fq(c)1440 2360 y Fo(1)1474 2351 y Fq(;:::)o(;c)1604 2362 y Fg(N)1679 2326 y Fw(\))29 b Fr(!)g Fu(K)1965 2290 y Fn(C)t Fp(\()p Fq(A)2086 2266 y Ff(0)2109 2290 y Fp(\))2141 2326 y Fw(,)k(where)i Fu(A)2557 2290 y Fn(0)2610 2326 y Fw(=)29 b Fu(\026)2774 2341 y Fq(k)2817 2326 y Fw(\()p Fu(A)p Fw(\).)46 b(W)-8 b(e)34 b(no)m(w)g(de\014ne)94 2459 y(a)40 b(p)s(olynomial)c(function)j Fu(F)53 b Fw(:)39 b Fu(K)1355 2423 y Fn(C)t Fp(\()p Fq(A)p Fp(\))1548 2459 y Fr(!)g Fu(K)47 b Fw(and)39 b(a)h(regular)e(map)h Fu(G)g Fw(:)h Fu(U)10 b Fw(\()p Fu(F)k Fw(\))40 b Fr(!)f Fu(K)3344 2423 y Fn(C)t Fp(\()p Fq(A)3465 2399 y Ff(0)3487 2423 y Fp(\))3558 2459 y Fw(b)m(y)94 2575 y(setting)94 2741 y(\(7.6\))1195 2741 y SDict begin H.S end 1195 2741 a 1195 2741 a SDict begin 14 H.A end 1195 2741 a 1195 2741 a SDict begin [ /View [/XYZ H.V] /Dest (equation.7.6) cvn H.B /DEST pdfmark end 1195 2741 a Fu(F)i Fw(=)28 b Fu(D)1487 2700 y Fq(d)1523 2709 y Fo(1)1558 2700 y Fq(;:::)n(;d)1692 2711 y Fg(N)1484 2765 y Fq(c)1515 2774 y Fo(1)1549 2765 y Fq(;:::)o(;c)1679 2776 y Fg(N)1776 2741 y Fr(\016)1867 2716 y Fm(e)1848 2741 y Fu(G;)114 b(G)28 b Fw(=)f Fu(H)j Fr(\016)2476 2716 y Fm(e)2457 2741 y Fu(G:)94 2898 y Fw(T)-8 b(o)34 b(\014nish)g(the)g(argumen)m(t)f(for)g Fu(`)c Fw(=)g(1,)34 b(it)f(remains)f(to)h(sho)m(w)i(that)e Fu(F)48 b Fw(is)33 b(not)g(iden)m(tically)f(equal)94 3014 y(to)i(zero.)48 b(But)34 b(this)f(is)g(clear)h(from)e(the)j (de\014nitions)e(\()2095 3014 y SDict begin H.S end 2095 3014 a Fw(4.14)2268 2950 y SDict begin H.R end 2268 2950 a 2268 3014 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.4.14) cvn H.B /ANN pdfmark end 2268 3014 a Fw(\))h(and)g(\()2569 3014 y SDict begin H.S end 2569 3014 a Fw(5.8)2694 2950 y SDict begin H.R end 2694 2950 a 2694 3014 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.8) cvn H.B /ANN pdfmark end 2694 3014 a Fw(\),)g(since)g(the)g(orien)m(ted)g(2-)94 3144 y(cycles)d(in)504 3119 y Fm(e)482 3144 y Fu(A)e Fw(\(up)h(to)f(cyclical)f(equiv)-5 b(alence\))30 b(are)f(of)g(the)h (form)e Fu(c)p Fw([)p Fu(ba)p Fw(])i(and)g(so)f(are)h(in)e(a)i (bijection)94 3260 y(with)j(the)g(orien)m(ted)f(3-cycles)h Fu(cba)g Fw(in)f Fu(A)h Fw(that)f(pass)i(through)e Fu(k)s Fw(.)194 3377 y(No)m(w)g(assume)g(that)g Fu(`)27 b Fr(\025)h Fw(2,)k(and)f(that)h(our)f(assertion)h(holds)f(if)f(w)m(e)j(replace)e Fu(`)h Fw(b)m(y)g Fu(`)20 b Fr(\000)g Fw(1.)43 b(Let)94 3493 y Fu(A)167 3508 y Fp(1)235 3493 y Fw(=)27 b Fu(\026)397 3508 y Fq(k)434 3517 y Fo(1)472 3493 y Fw(\()p Fu(A)p Fw(\),)f(so)e Fu(A)858 3457 y Fn(0)909 3493 y Fw(=)k Fu(\026)1072 3508 y Fq(k)1109 3520 y Fg(`)1159 3493 y Fr(\001)17 b(\001)g(\001)e Fu(\026)1351 3508 y Fq(k)1388 3517 y Fo(2)1426 3493 y Fw(\()p Fu(A)1537 3508 y Fp(1)1577 3493 y Fw(\).)40 b(By)25 b(the)f(inductiv)m(e)g(assumption,)h(there)g (exist)f(a)g(non-)94 3615 y(zero)36 b(p)s(olynomial)c(function)j Fu(F)1272 3579 y Fn(0)1328 3615 y Fw(:)e Fu(K)1478 3579 y Fn(C)t Fp(\()p Fq(A)1599 3588 y Fo(1)1634 3579 y Fp(\))1698 3615 y Fr(!)f Fu(K)43 b Fw(and)35 b(a)h(regular)e(map)h Fu(G)2863 3579 y Fn(0)2919 3615 y Fw(:)e Fu(U)10 b Fw(\()p Fu(F)3170 3579 y Fn(0)3193 3615 y Fw(\))33 b Fr(!)f Fu(K)3486 3579 y Fn(C)t Fp(\()p Fq(A)3607 3556 y Ff(0)3630 3579 y Fp(\))94 3731 y Fw(suc)m(h)k(that,)f(for)f(an)m(y)g(QP)h(\()p Fu(A)1181 3746 y Fp(1)1220 3731 y Fu(;)17 b(S)1324 3746 y Fp(1)1364 3731 y Fw(\))34 b(with)g Fu(S)1720 3746 y Fp(1)1790 3731 y Fr(2)d Fu(U)10 b Fw(\()p Fu(F)2078 3695 y Fn(0)2101 3731 y Fw(\),)35 b(the)f(QP)h Fu(\026)2606 3746 y Fq(k)2643 3758 y Fg(`)2693 3731 y Fr(\001)17 b(\001)g(\001)e Fu(\026)2885 3746 y Fq(k)2922 3755 y Fo(2)2960 3731 y Fw(\()p Fu(A)3071 3746 y Fp(1)3111 3731 y Fu(;)i(S)3215 3746 y Fp(1)3254 3731 y Fw(\))34 b(is)g(righ)m(t-)94 3848 y(equiv)-5 b(alen)m(t)33 b(to)f(\()p Fu(A)788 3811 y Fn(0)812 3848 y Fu(;)17 b(G)933 3811 y Fn(0)955 3848 y Fw(\()p Fu(S)1053 3863 y Fp(1)1093 3848 y Fw(\)\).)43 b(Also)32 b(b)m(y)i(the)f(already)f(established)h(case)g Fu(`)28 b Fw(=)f(1,)33 b(there)g(exists)h(a)94 3967 y(non-zero)g(p)s (olynomial)29 b(function)k Fu(F)1455 3931 y Fn(00)1525 3967 y Fw(:)c Fu(K)1671 3931 y Fn(C)t Fp(\()p Fq(A)1792 3940 y Fo(1)1827 3931 y Fp(\))1887 3967 y Fr(!)f Fu(K)40 b Fw(suc)m(h)35 b(that,)e(for)f(an)m(y)i(QP)f(\()p Fu(A)3218 3982 y Fp(1)3258 3967 y Fu(;)17 b(S)3362 3982 y Fp(1)3401 3967 y Fw(\))33 b(with)94 4083 y Fu(S)154 4098 y Fp(1)230 4083 y Fr(2)k Fu(U)10 b Fw(\()p Fu(F)524 4047 y Fn(00)567 4083 y Fw(\),)39 b(the)f(QP)g Fu(\026)1083 4098 y Fq(k)1120 4107 y Fo(1)1158 4083 y Fw(\()p Fu(A)1269 4098 y Fp(1)1308 4083 y Fu(;)17 b(S)1412 4098 y Fp(1)1451 4083 y Fw(\))38 b(is)f(2-acyclic,)h(hence)h(is)e(righ)m(t-equiv)-5 b(alen)m(t)36 b(to)i(some)f(QP)94 4199 y(on)c Fu(A)p Fw(.)44 b(Since)33 b(the)g(base)h(\014eld)e Fu(K)40 b Fw(is)32 b(assumed)i(to)e(b)s(e)h (in\014nite,)f(w)m(e)i(ha)m(v)m(e)g Fu(U)10 b Fw(\()p Fu(F)3009 4163 y Fn(0)3032 4199 y Fw(\))22 b Fr(\\)h Fu(U)10 b Fw(\()p Fu(F)3372 4163 y Fn(00)3415 4199 y Fw(\))28 b Fr(6)p Fw(=)g Fr(;)p Fw(.)94 4333 y(Cho)s(ose)38 b Fu(S)504 4282 y Fp(\(0\))498 4358 y(1)632 4333 y Fr(2)d Fu(U)10 b Fw(\()p Fu(F)924 4297 y Fn(0)947 4333 y Fw(\))25 b Fr(\\)g Fu(U)10 b Fw(\()p Fu(F)1292 4297 y Fn(00)1335 4333 y Fw(\),)38 b(and)e(let)g(\()p Fu(A;)17 b(S)1991 4348 y Fp(0)2030 4333 y Fw(\))34 b(=)h Fu(\026)2272 4348 y Fq(k)2314 4333 y Fw(\()p Fu(A)2425 4348 y Fp(1)2464 4333 y Fu(;)17 b(S)2574 4282 y Fp(\(0\))2568 4358 y(1)2668 4333 y Fw(\).)55 b(By)37 b(Theorem)3361 4333 y SDict begin H.S end 3361 4333 a Fw(5.7)3486 4269 y SDict begin H.R end 3486 4269 a 3486 4333 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.5.7) cvn H.B /ANN pdfmark end 3486 4333 a Fw(,)g(w)m(e)94 4467 y(ha)m(v)m(e)d Fu(\026)378 4482 y Fq(k)421 4467 y Fw(\()p Fu(A;)17 b(S)636 4482 y Fp(0)675 4467 y Fw(\))27 b(=)h(\()p Fu(A)955 4482 y Fp(1)995 4467 y Fu(;)17 b(S)1105 4416 y Fp(\(0\))1099 4492 y(1)1198 4467 y Fw(\).)44 b(By)33 b(the)g(ab)s(o)m(v)m(e)g(argumen)m(t)f(for)g Fu(`)c Fw(=)f(1,)33 b(there)g(exist)g(a)f(nonzero)94 4586 y(p)s(olynomial)43 b(function)i Fu(F)1070 4601 y Fp(1)1159 4586 y Fw(:)50 b Fu(K)1326 4550 y Fn(C)t Fp(\()p Fq(A)p Fp(\))1529 4586 y Fr(!)g Fu(K)j Fw(and)46 b(a)f(regular)g(map)g Fu(G)2763 4601 y Fp(1)2852 4586 y Fw(:)50 b Fu(U)10 b Fw(\()p Fu(F)3106 4601 y Fp(1)3146 4586 y Fw(\))50 b Fr(!)g Fu(K)3474 4550 y Fn(C)t Fp(\()p Fq(A)3595 4559 y Fo(1)3630 4550 y Fp(\))94 4703 y Fw(\(of)37 b(the)h(t)m(yp)s(e)g(\() 683 4703 y SDict begin H.S end 683 4703 a Fw(7.6)807 4638 y SDict begin H.R end 807 4638 a 807 4703 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.7.6) cvn H.B /ANN pdfmark end 807 4703 a Fw(\)\))f(suc)m(h)i(that)e Fu(\026)1420 4718 y Fq(k)1462 4703 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\))35 b(=)g(\()p Fu(A)1978 4718 y Fp(1)2018 4703 y Fu(;)17 b(G)2139 4718 y Fp(1)2178 4703 y Fw(\()p Fu(S)6 b Fw(\)\))37 b(for)f Fu(S)41 b Fr(2)36 b Fu(U)10 b Fw(\()p Fu(F)2928 4718 y Fp(1)2968 4703 y Fw(\).)58 b(In)37 b(particular,)94 4836 y(w)m(e)j(ha)m(v)m(e)g Fu(G)552 4851 y Fp(1)591 4836 y Fw(\()p Fu(S)689 4851 y Fp(0)729 4836 y Fw(\))d(=)h Fu(S)984 4786 y Fp(\(0\))978 4861 y(1)1116 4836 y Fw(implying)e(that)i Fu(F)1820 4800 y Fn(0)1870 4836 y Fr(\016)25 b Fu(G)2022 4851 y Fp(1)2100 4836 y Fw(is)38 b(a)h(nonzero)g(p)s(olynomial)34 b(function)k(on)94 4956 y Fu(K)184 4919 y Fn(C)t Fp(\()p Fq(A)p Fp(\))337 4956 y Fw(.)43 b(It)31 b(follo)m(ws)e(that)h(the)h (nonzero)g(p)s(olynomial)26 b(function)k Fu(F)14 b Fw(\()p Fu(S)6 b Fw(\))27 b(=)h Fu(F)2859 4971 y Fp(1)2898 4956 y Fw(\()p Fu(S)6 b Fw(\))p Fu(F)3117 4919 y Fn(0)3140 4956 y Fw(\()p Fu(G)3255 4971 y Fp(1)3294 4956 y Fw(\()p Fu(S)g Fw(\)\))30 b(and)94 5076 y(the)39 b(regular)d(map)h Fu(G)g Fw(=)f Fu(G)1128 5040 y Fn(0)1177 5076 y Fr(\016)26 b Fu(G)1330 5091 y Fp(1)1405 5076 y Fw(:)37 b Fu(U)10 b Fw(\()p Fu(F)k Fw(\))37 b Fr(!)f Fu(K)1961 5040 y Fn(C)t Fp(\()p Fq(A)2082 5017 y Ff(0)2105 5040 y Fp(\))2174 5076 y Fw(are)i(w)m(ell-de\014ned)g(and)g(satisfy)f(all)f(the)94 5192 y(required)e(conditions.)42 b(This)33 b(completes)f(the)h(pro)s (of)f(of)g(Prop)s(osition)2713 5192 y SDict begin H.S end 2713 5192 a Fw(7.3)2838 5128 y SDict begin H.R end 2838 5128 a 2838 5192 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.7.3) cvn H.B /ANN pdfmark end 2838 5192 a Fw(.)719 b Fh(\003)94 5262 y SDict begin H.S end 94 5262 a 94 5262 a SDict begin 14 H.A end 94 5262 a 94 5262 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.7.4) cvn H.B /DEST pdfmark end 94 5262 a 85 x FA(Corollary)35 b(7.4.)40 b Fv(F)-7 b(or)31 b(every)h Fw(2)p Fv(-acyclic)g(arr)-5 b(ow)32 b(sp)-5 b(an)32 b Fu(A)p Fv(,)g(ther)-5 b(e)33 b(exists)f(a)g(c)-5 b(ountable)32 b(family)g Fr(F)94 5464 y Fv(of)j(nonzer)-5 b(o)34 b(p)-5 b(olynomial)33 b(functions)h(on)h Fu(K)1717 5428 y Fn(C)t Fp(\()p Fq(A)p Fp(\))1904 5464 y Fv(such)g(that)g(the)g (QP)f Fw(\()p Fu(A;)17 b(S)6 b Fw(\))34 b Fv(is)h(nonde)-5 b(gener)g(ate)94 5580 y(whenever)42 b Fu(S)48 b Fr(2)743 5505 y Fm(T)826 5609 y Fq(F)10 b Fn(2F)1006 5580 y Fu(U)g Fw(\()p Fu(F)k Fw(\))p Fv(.)68 b(In)42 b(p)-5 b(articular,)44 b(if)f(the)f(b)-5 b(ase)42 b(\014eld)g Fu(K)50 b Fv(is)42 b(unc)-5 b(ountable,)45 b(then)94 5696 y(ther)-5 b(e)35 b(exists)g(a)g(nonde)-5 b(gener)g(ate)33 b(QP)i(on)f Fu(A)p Fv(.)p eop end end %%Page: 32 32 TeXDict begin HPSdict begin 32 31 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.32) cvn H.B /DEST pdfmark end 94 218 a Fs(32)492 b(HARM)25 b(DERKSEN,)f(JERZY)i(WEYMAN,)f(AND)f(ANDREI)g(ZELEVINSKY)94 417 y Fv(Pr)-5 b(o)g(of.)42 b Fw(By)e(Prop)s(osition)1085 417 y SDict begin H.S end 1085 417 a Fw(7.3)1210 353 y SDict begin H.R end 1210 353 a 1210 417 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.7.3) cvn H.B /ANN pdfmark end 1210 417 a Fw(,)h(for)e(ev)m(ery)j(sequence)g Fu(k)2161 432 y Fp(1)2201 417 y Fu(;)17 b(:)g(:)g(:)e(;)i(k)2470 432 y Fq(`)2542 417 y Fw(as)40 b(in)f(De\014nition)3251 417 y SDict begin H.S end 3251 417 a Fw(7.2)3376 353 y SDict begin H.R end 3376 353 a 3376 417 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.7.2) cvn H.B /ANN pdfmark end 3376 417 a Fw(,)j(there)94 533 y(exists)47 b(a)f(nonzero)h(p)s (olynomial)42 b(function)k Fu(F)1825 548 y Fq(k)1862 557 y Fo(1)1897 548 y Fq(;:::)n(;k)2032 560 y Fg(`)2112 533 y Fw(on)g Fu(K)2351 497 y Fn(C)t Fp(\()p Fq(A)p Fp(\))2550 533 y Fw(suc)m(h)h(that)f(a)g(QP)g(\()p Fu(A;)17 b(S)6 b Fw(\))46 b(is)94 649 y(\()p Fu(k)183 664 y Fq(`)216 649 y Fu(;)17 b(:)g(:)g(:)f(;)h(k)486 664 y Fp(1)525 649 y Fw(\)-nondegenerate)26 b(for)e Fu(S)34 b Fr(2)28 b Fu(U)10 b Fw(\()p Fu(F)1739 664 y Fq(k)1776 673 y Fo(1)1811 664 y Fq(;:::)n(;k)1946 676 y Fg(`)1980 649 y Fw(\).)40 b(These)27 b(functions)e(form)f(a)h(desired)g(coun)m(t-)94 766 y(able)32 b(family)e Fr(F)10 b Fw(.)194 882 y(It)45 b(remains)g(to)g(pro)m(v)m(e)h(that)1320 807 y Fm(T)1403 911 y Fq(F)10 b Fn(2F)1583 882 y Fu(U)g Fw(\()p Fu(F)k Fw(\))49 b Fr(6)p Fw(=)g Fr(;)c Fw(pro)m(vided)h Fu(K)52 b Fw(is)45 b(uncoun)m(table.)82 b(If)46 b Fu(A)f Fw(is)94 1009 y(acyclic,)36 b(i.e.,)g(has)g(no)g(orien)m(ted)f(cycles,)j(then)e Fu(K)1951 973 y Fn(C)t Fp(\()p Fq(A)p Fp(\))2136 1009 y Fw(=)d Fr(f)p Fw(0)p Fr(g)p Fw(,)j(and)f(eac)m(h)i(of)e(the)h (functions)f(in)94 1126 y Fr(F)48 b Fw(is)37 b(just)h(a)g(nonzero)g (constan)m(t,)i(so)e(there)h(is)e(nothing)g(to)h(pro)m(v)m(e;)j(no)d (assumption)f(on)h Fu(K)45 b Fw(is)94 1242 y(needed)27 b(here.)42 b(If)25 b Fu(A)g Fw(has)g(at)g(least)f(one)i(orien)m(ted)f (cycle)g(then)h(the)f(set)h Fr(C)6 b Fw(\()p Fu(A)p Fw(\))25 b(is)g(coun)m(table)g(\(recall)94 1361 y(that)h(it)f(consists)i(of)f (cyclic)g(paths)g(up)h(to)f(cyclical)e(equiv)-5 b(alence\).)42 b(Th)m(us,)29 b(w)m(e)e(can)f(realize)f Fu(K)3508 1325 y Fn(C)t Fp(\()p Fq(A)p Fp(\))94 1477 y Fw(as)30 b(the)g(p)s(olynomial) c(ring)i Fu(K)7 b Fw([)p Fu(X)1272 1492 y Fp(1)1312 1477 y Fu(;)17 b(X)1437 1492 y Fp(2)1476 1477 y Fu(;)g(:)g(:)g(:)e Fw(])30 b(in)e(coun)m(tably)i(man)m(y)f(indeterminates.)42 b(Since)29 b Fu(K)37 b Fw(is)94 1594 y(uncoun)m(table,)g(w)m(e)g(can)e (c)m(ho)s(ose)i Fu(x)1371 1609 y Fp(1)1446 1594 y Fw(so)e(that)h Fu(F)14 b Fw(\()p Fu(x)1953 1609 y Fp(1)1992 1594 y Fw(\))32 b Fr(6)p Fw(=)h(0)i(for)g(all)e Fu(F)46 b Fr(2)33 b(F)h(\\)24 b Fu(K)7 b Fw([)p Fu(X)3147 1609 y Fp(1)3187 1594 y Fw(].)52 b(Then)36 b(w)m(e)94 1710 y(c)m(ho)s(ose)f Fu(x)460 1725 y Fp(2)533 1710 y Fw(so)e(that)g Fu(F)14 b Fw(\()p Fu(x)1035 1725 y Fp(1)1075 1710 y Fu(;)j(x)1174 1725 y Fp(2)1213 1710 y Fw(\))29 b Fr(6)p Fw(=)g(0)k(for)f(all)g Fu(F)42 b Fr(2)29 b(F)k(\\)23 b Fu(K)7 b Fw([)p Fu(X)2345 1725 y Fp(1)2384 1710 y Fu(;)17 b(X)2509 1725 y Fp(2)2548 1710 y Fw(].)46 b(Con)m(tin)m(uing)33 b(lik)m(e)f(this,)h(w)m(e)94 1826 y(\014nd)h(a)e(sequence)j Fu(x)830 1841 y Fp(1)870 1826 y Fu(;)17 b(x)969 1841 y Fp(2)1009 1826 y Fu(;)g(:)g(:)g(:)48 b Fw(suc)m(h)34 b(that)e Fu(F)14 b Fw(\()p Fu(x)1817 1841 y Fp(1)1857 1826 y Fu(;)j(x)1956 1841 y Fp(2)1995 1826 y Fu(;)g(:)g(:)g(:)f Fw(\))28 b Fr(6)p Fw(=)f(0)33 b(for)f(all)e Fu(F)42 b Fr(2)28 b(F)10 b Fw(.)571 b Fh(\003)94 1932 y SDict begin H.S end 94 1932 a 94 1932 a SDict begin 14 H.A end 94 1932 a 94 1932 a SDict begin [ /View [/XYZ H.V] /Dest (section.8) cvn H.B /DEST pdfmark end 94 1932 a 1562 2024 a Fw(8.)49 b Fx(Rigid)37 b(QPs)94 2101 y SDict begin H.S end 94 2101 a 94 2101 a SDict begin 14 H.A end 94 2101 a 94 2101 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.8.1) cvn H.B /DEST pdfmark end 94 2101 a 97 x FA(Prop)s(osition)f(8.1.)42 b Fv(Every)35 b(rigid)f(r)-5 b(e)g(duc)g(e)g(d)35 b(QP)f Fw(\()p Fu(A;)17 b(S)6 b Fw(\))35 b Fv(is)f Fw(2)p Fv(-acyclic.)94 2358 y(Pr)-5 b(o)g(of.)42 b Fw(Supp)s(ose)24 b(for)e(the)h(sak)m(e)i(of)d(con)m (tradiction)g(that)h(for)f(some)h Fu(i)28 b Fr(6)p Fw(=)f Fu(j)i Fw(b)s(oth)23 b(comp)s(onen)m(ts)g Fu(A)3581 2373 y Fq(i;j)94 2474 y Fw(and)k Fu(A)351 2489 y Fq(j;i)454 2474 y Fw(are)f(non-zero.)42 b(Cho)s(ose)27 b(non-zero)g(elemen)m(ts)f Fu(a)i Fr(2)g Fu(A)2403 2489 y Fq(i;j)2510 2474 y Fw(and)f Fu(b)h Fr(2)g Fu(A)2930 2489 y Fq(j;i)3006 2474 y Fw(.)42 b(Remem)m(b)s(ering)94 2591 y(the)25 b(de\014nition)f(of)g(the)h (Jacobian)e(ideal)g(\(see)j(De\014nition)2200 2591 y SDict begin H.S end 2200 2591 a Fw(3.1)2325 2526 y SDict begin H.R end 2325 2526 a 2325 2591 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.3.1) cvn H.B /ANN pdfmark end 2325 2591 a Fw(\),)g(it)d(is)h(easy)i(to)e(see)i(that)e(the)h(cyclic) 94 2707 y(part)37 b(of)f Fu(J)9 b Fw(\()p Fu(S)d Fw(\))36 b(is)f(con)m(tained)i(in)f Fl(m)p Fw(\()p Fu(A)p Fw(\))1556 2671 y Fp(3)1556 2731 y(cyc)1660 2707 y Fw(.)55 b(It)36 b(follo)m(ws)f(that)h Fu(ab)h Fw(is)f Fv(not)46 b Fw(cyclically)35 b(equiv)-5 b(alen)m(t)94 2827 y(to)35 b(an)h(elemen)m(t)f(of)g Fu(J)9 b Fw(\()p Fu(S)d Fw(\),)35 b(hence)i(it)d(giv)m(es)i(rise)f(to)g (a)g(non-zero)h(elemen)m(t)f(of)g(the)g(deformation)94 2944 y(space)f(Def)7 b(\()p Fu(A;)17 b(S)6 b Fw(\),)32 b(giving)f(the)i(desired)g(con)m(tradiction.)1378 b Fh(\003)194 3110 y Fw(Com)m(bining)31 b(Prop)s(osition)1215 3110 y SDict begin H.S end 1215 3110 a Fw(8.1)1339 3045 y SDict begin H.R end 1339 3045 a 1339 3110 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.8.1) cvn H.B /ANN pdfmark end 1339 3110 a 33 w Fw(with)h(Corollary)2025 3110 y SDict begin H.S end 2025 3110 a Fw(6.11)2198 3045 y SDict begin H.R end 2198 3045 a 2198 3110 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.6.11) cvn H.B /ANN pdfmark end 2198 3110 a Fw(,)h(w)m(e)h(obtain)d(the)i(follo)m(wing)d(result.)94 3129 y SDict begin H.S end 94 3129 a 94 3129 a SDict begin 14 H.A end 94 3129 a 94 3129 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.8.2) cvn H.B /DEST pdfmark end 94 3129 a 141 x FA(Corollary)38 b(8.2.)k Fv(A)n(ny)35 b(rigid)f(QP)h(is)f (nonde)-5 b(gener)g(ate.)194 3430 y Fw(Let)33 b(us)g(no)m(w)g(giv)m(e)g (some)f(examples.)94 3449 y SDict begin H.S end 94 3449 a 94 3449 a SDict begin 14 H.A end 94 3449 a 94 3449 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.8.3) cvn H.B /DEST pdfmark end 94 3449 a 141 x FA(Example)e(8.3.)37 b Fw(Recall)24 b(that)i(a)g(sk)m(ew-symmetric)h(in)m(teger)g(matrix)d Fu(B)32 b Fw(is)25 b Fv(acyclic)32 b Fw(if)25 b(the)i(corre-)94 3706 y(sp)s(onding)d(directed)f(graph)h(\(with)f(an)g(arro)m(w)h Fu(i)k Fr(!)f Fu(j)i Fw(asso)s(ciated)24 b(with)f(eac)m(h)h(en)m(try)h Fu(b)3198 3721 y Fq(i;j)3306 3706 y Fu(>)j Fw(0\))23 b(has)94 3823 y(no)h(orien)m(ted)g(cycles.)41 b(If)24 b(the)g(matrix)e Fu(B)5 b Fw(\()p Fu(A)p Fw(\))24 b(giv)m(en)g(b)m(y)h (\()2122 3823 y SDict begin H.S end 2122 3823 a Fw(7.1)2246 3758 y SDict begin H.R end 2246 3758 a 2246 3823 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.7.1) cvn H.B /ANN pdfmark end 2246 3823 a Fw(\))f(is)f(acyclic,)i(then)f Fu(R)q Fr(hh)p Fu(A)p Fr(ii)3251 3838 y Fp(cyc)3382 3823 y Fw(=)k Fr(f)p Fw(0)p Fr(g)p Fw(,)94 3939 y(and)33 b(so)g(the)g(only)f (QP)h(asso)s(ciated)f(with)g Fu(A)h Fw(is)f(\()p Fu(A;)17 b Fw(0\),)32 b(whic)m(h)h(is)f(clearly)g(rigid.)194 4055 y(No)m(w)45 b(supp)s(ose)h(that)e Fu(A)g Fw(is)g(2-acyclic,)i(and)e (that)g Fu(B)5 b Fw(\()p Fu(A)p Fw(\))45 b(is)e(not)i(necessarily)f (acyclic)g(but)94 4171 y(is)38 b(m)m(utation)f(equiv)-5 b(alen)m(t)39 b(to)f(an)g(acyclic)g(matrix)f(\(i.e.,)j(can)f(b)s(e)f (transformed)g(to)g(an)h(acyclic)94 4288 y(matrix)33 b(b)m(y)i(a)e(sequence)k(of)d(m)m(utations\).)46 b(As)34 b(a)g(consequence)j(of)d(Corollary)3014 4288 y SDict begin H.S end 3014 4288 a Fw(6.11)3187 4223 y SDict begin H.R end 3187 4223 a 3187 4288 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.6.11) cvn H.B /ANN pdfmark end 3187 4288 a 34 w Fw(and)g(Theo-)94 4404 y(rem)287 4404 y SDict begin H.S end 287 4404 a Fw(5.7)412 4340 y SDict begin H.R end 412 4340 a 412 4404 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.5.7) cvn H.B /ANN pdfmark end 412 4404 a Fw(,)d(there)g(exists)g(a)g(p)s(oten)m(tial)d Fu(S)36 b Fw(suc)m(h)c(that)f(\()p Fu(A;)17 b(S)6 b Fw(\))30 b(is)f(a)i(rigid)d(reduced)k(QP;)f(moreo)m(v)m(er,)94 4520 y(\()p Fu(A;)17 b(S)6 b Fw(\))32 b(is)h(unique)g(up)g(to)f(righ)m (t-equiv)-5 b(alences.)94 4589 y SDict begin H.S end 94 4589 a 94 4589 a SDict begin 14 H.A end 94 4589 a 94 4589 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.8.4) cvn H.B /DEST pdfmark end 94 4589 a 91 x FA(Example)44 b(8.4.)g Fw(F)-8 b(or)38 b Fu(A)g Fw(arbitrary)-8 b(,)39 b(the)g(deformation)d (space)k(of)e(a)g(QP)g(\()p Fu(A;)17 b Fw(0\))38 b(is)g(naturally)94 4796 y(iden)m(ti\014ed)g(with)g(the)h(cyclic)e(part)h(of)g Fl(m)p Fw(\()p Fu(A)p Fw(\),)i(hence)f(it)e(is)h (in\014nite-dimensional)c(pro)m(vided)k Fu(A)94 4913 y Fw(has)33 b(at)g(least)f(one)h(orien)m(ted)f(cycle.)94 4976 y SDict begin H.S end 94 4976 a 94 4976 a SDict begin 14 H.A end 94 4976 a 94 4976 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.8.5) cvn H.B /DEST pdfmark end 94 4976 a 97 x FA(Example)39 b(8.5)h Fw(\(Cyclic)35 b(triangle\))p FA(.)40 b Fw(Let)35 b Fu(Q)g Fw(b)s(e)g(the)g(quiv)m(er)g(with)g(three) g(v)m(ertices)h(1)p Fu(;)17 b Fw(2)p Fu(;)g Fw(3)33 b(and)94 5189 y(three)h(arro)m(ws)f Fu(a)28 b Fw(:)g(1)f Fr(!)g Fw(2,)33 b Fu(b)28 b Fw(:)g(2)f Fr(!)g Fw(3)33 b(and)f Fu(c)c Fw(:)g(3)f Fr(!)h Fw(1:)1854 5353 y(2)2060 5455 y Fq(b)2131 5587 y Fk(\036)p Fj(\036)2102 5556 y Fi(=)2074 5526 y(=)2045 5497 y(=)2017 5467 y(=)1989 5437 y(=)1960 5407 y(=)1932 5377 y(=)1547 5675 y Fw(1)1658 5455 y Fq(a)1825 5377 y Fk(@)p Fj(@)1796 5408 y Fi(\001)1768 5437 y(\001)1739 5467 y(\001)1711 5497 y(\001)1683 5527 y(\001)1654 5557 y(\001)1626 5587 y(\001)2160 5675 y Fw(3)1863 5700 y Fq(c)1625 5643 y Fk(o)p Fj(o)p 1625 5644 506 4 v eop end end %%Page: 33 33 TeXDict begin HPSdict begin 33 32 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.33) cvn H.B /DEST pdfmark end 94 218 a 1179 w Fs(QUIVERS)31 b(WITH)i(POTENTIALS)g(I)1102 b(33)94 417 y Fw(An)34 b(arbitrary)e(p)s(oten)m(tial)g Fu(S)39 b Fw(is)33 b(cyclically)e(equiv)-5 b(alen)m(t)33 b(to)g(the)h(one)f(of) g(the)h(form)e Fu(S)i Fw(=)29 b Fu(F)14 b Fw(\()p Fu(cba)p Fw(\),)94 533 y(where)49 b Fu(F)65 b Fr(2)53 b Fu(K)7 b Fw([[)p Fu(t)p Fw(]])47 b(is)g(a)f(formal)f(p)s(o)m(w)m(er)i(series.) 87 b(The)48 b(deformation)d(space)j(Def)7 b(\()p Fu(A;)17 b(S)6 b Fw(\))46 b(is)94 649 y(naturally)32 b(isomorphic)g(to)h(the)h (quotien)m(t)f(space)i(of)e Fu(tK)7 b Fw([[)p Fu(t)p Fw(]])34 b(mo)s(dulo)e(the)h(ideal)f(generated)i(b)m(y)94 766 y Fu(tdF)8 b(=dt)p Fw(.)43 b(If)28 b(c)m(har)q Fu(K)34 b Fw(=)28 b(0,)h(and)g Fu(n)f Fr(\025)g Fw(1)g(is)g(the)h(smallest)f (exp)s(onen)m(t)i(suc)m(h)g(that)e Fu(t)3015 730 y Fq(n)3091 766 y Fw(app)s(ears)h(in)f Fu(F)14 b Fw(,)94 882 y(then)34 b(dim)15 b(Def)6 b(\()p Fu(A;)17 b(S)6 b Fw(\))28 b(=)f Fu(n)c Fr(\000)f Fw(1.)43 b(In)33 b(particular,)e(\()p Fu(A;)17 b(S)6 b Fw(\))32 b(is)g(rigid)f(if)g(and)i(only)f(if)g Fu(n)27 b Fw(=)h(1.)194 1012 y(No)m(w)k(let)f(us)h(consider)g(the)f(QP) h(\()1458 987 y Fm(e)1436 1012 y Fu(A;)1566 987 y Fm(e)1553 1012 y Fu(S)6 b Fw(\))27 b(=)32 b Fm(e)-59 b Fu(\026)1847 1027 y Fp(2)1886 1012 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\);)31 b(in)g(view)h(of)f(\()2688 1012 y SDict begin H.S end 2688 1012 a Fw(5.6)2812 948 y SDict begin H.R end 2812 948 a 2812 1012 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.6) cvn H.B /ANN pdfmark end 2812 1012 a Fw(\),)h(\()2947 1012 y SDict begin H.S end 2947 1012 a Fw(5.7)3071 948 y SDict begin H.R end 3071 948 a 3071 1012 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.7) cvn H.B /ANN pdfmark end 3071 1012 a Fw(\))g(and)f(\()3367 1012 y SDict begin H.S end 3367 1012 a Fw(5.8)3492 948 y SDict begin H.R end 3492 948 a 3492 1012 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.8) cvn H.B /ANN pdfmark end 3492 1012 a Fw(\),)3611 987 y Fm(e)3588 1012 y Fu(A)94 1128 y Fw(has)i(four)g(arro)m(ws)g Fu(a)835 1092 y Fq(?)874 1128 y Fu(;)17 b(b)959 1092 y Fq(?)999 1128 y Fu(;)g(c;)g Fw([)p Fu(ba)p Fw(],)33 b(and)1389 1300 y Fm(e)1376 1325 y Fu(S)g Fw(=)28 b Fu(F)14 b Fw(\()p Fu(c)p Fw([)p Fu(ba)p Fw(]\))22 b(+)g([)p Fu(ba)p Fw(])p Fu(a)2231 1284 y Fq(?)2272 1325 y Fu(b)2313 1284 y Fq(?)2353 1325 y Fu(:)94 1522 y Fw(Th)m(us,)42 b(if)c Fu(n)g Fr(\025)g Fw(2)h(then)g(\()1059 1496 y Fm(e)1037 1522 y Fu(A;)1167 1496 y Fm(e)1154 1522 y Fu(S)6 b Fw(\))38 b(is)g(reduced)i(and)f(so)g(is)f(equal)g(to)h Fu(\026)2643 1537 y Fp(2)2682 1522 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\))37 b(=)h(\()p 3130 1442 74 4 v Fu(A;)p 3247 1442 66 4 v 17 w(S)6 b Fw(\).)61 b(Since)94 1638 y Fu(\026)153 1653 y Fp(2)193 1638 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\))35 b(has)g(an)h(orien)m(ted)f(2-cycle)h(formed)f(b)m(y)h (the)g(arro)m(ws)g Fu(c)g Fw(and)f([)p Fu(ba)p Fw(],)i(the)f(m)m (utations)f(at)94 1754 y(v)m(ertices)40 b(1)d(and)h(3)g(cannot)g(b)s(e) g(applied.)58 b(W)-8 b(e)38 b(see)h(that)f(the)g(QP)g(\()p Fu(A;)17 b(F)d Fw(\()p Fu(cba)p Fw(\)\))38 b(is)f(degenerate)94 1870 y(for)c Fu(n)27 b Fr(\025)i Fw(2.)94 1957 y SDict begin H.S end 94 1957 a 94 1957 a SDict begin 14 H.A end 94 1957 a 94 1957 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.8.6) cvn H.B /DEST pdfmark end 94 1957 a 102 x FA(Example)36 b(8.6)g Fw(\(Double)31 b(cyclic)h(triangle\))p FA(.)39 b Fw(No)m(w)32 b(consider)g(the)g(quiv)m(er)g(with)g(three)g(v)m (ertices)94 2176 y(1)p Fu(;)17 b Fw(2)p Fu(;)g Fw(3)32 b(and)g(six)h(arro)m(ws)g Fu(a)1063 2191 y Fp(1)1103 2176 y Fu(;)17 b(a)1198 2191 y Fp(2)1265 2176 y Fw(:)28 b(1)f Fr(!)g Fw(2,)33 b Fu(b)1673 2191 y Fp(1)1713 2176 y Fu(;)17 b(b)1798 2191 y Fp(2)1865 2176 y Fw(:)28 b(2)f Fr(!)g Fw(3)33 b(and)f Fu(c)2436 2191 y Fp(1)2476 2176 y Fu(;)17 b(c)2562 2191 y Fp(2)2629 2176 y Fw(:)27 b(3)h Fr(!)f Fw(1:)1854 2365 y(2)2076 2444 y Fq(b)2106 2453 y Fo(1)2146 2584 y Fk(\036)p Fj(\036)2118 2554 y Fi(=)2089 2524 y(=)2061 2494 y(=)2033 2464 y(=)2004 2434 y(=)1976 2404 y(=)1947 2375 y(=)1922 2581 y Fq(b)1952 2590 y Fo(2)2115 2613 y Fk(\036)p Fj(\036)2087 2583 y Fi(=)2058 2553 y(=)2030 2523 y(=)2001 2493 y(=)1973 2464 y(=)1945 2434 y(=)1916 2404 y(=)1547 2687 y Fw(1)1608 2443 y Fq(a)1645 2452 y Fo(1)1809 2375 y Fk(@)p Fj(@)1780 2405 y Fi(\001)1752 2435 y(\001)1724 2464 y(\001)1695 2494 y(\001)1667 2524 y(\001)1638 2554 y(\001)1610 2584 y(\001)1770 2564 y Fq(a)1807 2573 y Fo(2)1840 2404 y Fk(@)p Fj(@)1812 2434 y Fi(\001)1783 2464 y(\001)1755 2494 y(\001)1727 2524 y(\001)1698 2553 y(\001)1670 2583 y(\001)1641 2613 y(\001)2160 2687 y Fw(3)1845 2595 y Fq(c)1876 2604 y Fo(2)1625 2633 y Fk(o)p Fj(o)p 1625 2635 506 4 v 1845 2734 a Fq(c)1876 2743 y Fo(1)1625 2676 y Fk(o)p Fj(o)p 1625 2678 V 194 2916 a Fw(An)m(y)h(p)s(oten)m(tial)d Fu(S)32 b Fw(on)27 b Fu(A)g Fw(is)f(cyclically)f(equiv)-5 b(alen)m(t)27 b(to)f(the)h(one)g(whose)h(degree)g(3)e(comp)s(onen)m(t)94 3032 y(b)s(elongs)g(to)f(the)h(8-dimensional)d(space)k Fu(A)1650 2996 y Fp(3)1650 3057 y(1)p Fq(;)p Fp(1)1772 3032 y Fw(=)g Fu(A)1948 3047 y Fp(1)p Fq(;)p Fp(3)2043 3032 y Fu(A)2116 3047 y Fp(3)p Fq(;)p Fp(2)2210 3032 y Fu(A)2283 3047 y Fp(2)p Fq(;)p Fp(1)2378 3032 y Fw(.)41 b(It)26 b(is)f(kno)m(wn)i(that)f(the)g(diagonal)94 3157 y(action)36 b(of)h(the)g(group)g Fu(GL)1103 3121 y Fp(3)1103 3181 y(2)1179 3157 y Fw(on)g Fu(K)1409 3121 y Fp(2)1473 3157 y Fr(\012)26 b Fu(K)1666 3121 y Fp(2)1731 3157 y Fr(\012)f Fu(K)1923 3121 y Fp(2)1999 3157 y Fw(has)38 b(sev)m(en)h(orbits,)e(see)h(e.g.,)g([)3152 3157 y SDict begin H.S end 3152 3157 a Fw(23)3250 3093 y SDict begin H.R end 3250 3093 a 3250 3157 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.gkz) cvn H.B /ANN pdfmark end 3250 3157 a Fw(,)g(Chapter)94 3273 y(14,)i(Example)e(4.5].)60 b(Th)m(us,)41 b(b)m(y)f(p)s(erforming)c(a)i(c)m(hange)h(of)f(arro)m(ws) h(automorphism,)f(w)m(e)h(can)94 3389 y(assume)27 b(that)f(the)h (degree)g(3)f(comp)s(onen)m(t)g(of)g Fu(S)32 b Fw(is)25 b(one)i(of)f(the)g(represen)m(tativ)m(es)j(of)c(these)j(orbits.)94 3505 y(An)41 b(easy)h(case-b)m(y-case)g(analysis)d(sho)m(ws)j(that)f (no)f(p)s(oten)m(tial)f(can)h(giv)m(e)g(rise)h(to)f(a)g(rigid)e(QP)94 3622 y(on)33 b Fu(A)p Fw(.)194 3738 y(F)-8 b(or)32 b(instance,)h(let)94 3920 y(\(8.1\))1453 3920 y SDict begin H.S end 1453 3920 a 1453 3920 a SDict begin 14 H.A end 1453 3920 a 1453 3920 a SDict begin [ /View [/XYZ H.V] /Dest (equation.8.1) cvn H.B /DEST pdfmark end 1453 3920 a Fu(S)g Fw(=)28 b Fu(c)1692 3935 y Fp(1)1731 3920 y Fu(b)1772 3935 y Fp(1)1812 3920 y Fu(a)1863 3935 y Fp(1)1925 3920 y Fw(+)22 b Fu(c)2065 3935 y Fp(2)2104 3920 y Fu(b)2145 3935 y Fp(2)2185 3920 y Fu(a)2236 3935 y Fp(2)2276 3920 y Fu(:)94 4103 y Fw(Then)34 b Fu(J)9 b Fw(\()p Fu(S)d Fw(\))33 b(is)f(the)h(closure)f(of)g(the)h (ideal)e(in)h Fu(R)q Fr(hh)p Fu(A)p Fr(ii)g Fw(generated)h(b)m(y)h(six) e(elemen)m(ts)1249 4285 y Fu(c)1291 4300 y Fp(1)1331 4285 y Fu(b)1372 4300 y Fp(1)1412 4285 y Fu(;)17 b(b)1497 4300 y Fp(1)1536 4285 y Fu(a)1587 4300 y Fp(1)1627 4285 y Fu(;)g(a)1722 4300 y Fp(1)1761 4285 y Fu(c)1803 4300 y Fp(1)1843 4285 y Fu(;)g(c)1929 4300 y Fp(2)1968 4285 y Fu(b)2009 4300 y Fp(2)2049 4285 y Fu(;)g(b)2134 4300 y Fp(2)2173 4285 y Fu(a)2224 4300 y Fp(2)2264 4285 y Fu(;)g(a)2359 4300 y Fp(2)2398 4285 y Fu(c)2440 4300 y Fp(2)2480 4285 y Fu(:)94 4468 y Fw(One)34 b(c)m(hec)m(ks)j(easily)32 b(that)i(the)g(image)d(of)i Fu(c)1691 4483 y Fp(1)1731 4468 y Fu(b)1772 4483 y Fp(2)1812 4468 y Fu(a)1863 4483 y Fp(1)1902 4468 y Fu(c)1944 4483 y Fp(2)1984 4468 y Fu(b)2025 4483 y Fp(1)2065 4468 y Fu(a)2116 4483 y Fp(2)2189 4468 y Fw(in)g(T)-8 b(r\()p Fr(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\)\))33 b(giv)m(es)h(rise)f(to)h(a)f(non-)94 4584 y(zero)39 b(elemen)m(t)f(of)g(the)g(deformation)e(space,)41 b(so)d(\()p Fu(A;)17 b(S)6 b Fw(\))38 b(is)f(not)h(rigid.)58 b(No)m(w)39 b(let)f(us)g(compute)94 4714 y Fu(\026)153 4729 y Fp(2)193 4714 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\).)43 b(Again)31 b(setting)h(\()1187 4689 y Fm(e)1164 4714 y Fu(A;)1294 4689 y Fm(e)1281 4714 y Fu(S)6 b Fw(\))28 b(=)j Fm(e)-59 b Fu(\026)1575 4729 y Fp(2)1614 4714 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\),)32 b(w)m(e)i(see)g(that)2467 4689 y Fm(e)2445 4714 y Fu(A)f Fw(has)g(ten)g(arro)m(ws)963 4897 y Fu(a)1014 4856 y Fq(?)1014 4921 y Fp(1)1054 4897 y Fu(;)17 b(a)1149 4856 y Fq(?)1149 4921 y Fp(2)1188 4897 y Fu(;)g(b)1273 4856 y Fq(?)1273 4921 y Fp(1)1313 4897 y Fu(;)g(b)1398 4856 y Fq(?)1398 4921 y Fp(2)1437 4897 y Fu(;)g(c)1523 4912 y Fp(1)1563 4897 y Fu(;)g(c)1649 4912 y Fp(2)1688 4897 y Fu(;)g Fw([)p Fu(b)1800 4912 y Fp(1)1839 4897 y Fu(a)1890 4912 y Fp(1)1930 4897 y Fw(])p Fu(;)g Fw([)p Fu(b)2069 4912 y Fp(1)2109 4897 y Fu(a)2160 4912 y Fp(2)2200 4897 y Fw(])p Fu(;)g Fw([)p Fu(b)2339 4912 y Fp(2)2378 4897 y Fu(a)2429 4912 y Fp(1)2469 4897 y Fw(])p Fu(;)g Fw([)p Fu(b)2608 4912 y Fp(2)2648 4897 y Fu(a)2699 4912 y Fp(2)2738 4897 y Fw(])p Fu(;)94 5079 y Fw(and)1077 5265 y Fm(e)1064 5290 y Fu(S)33 b Fw(=)28 b Fu(c)1303 5305 y Fp(1)1342 5290 y Fw([)p Fu(b)1410 5305 y Fp(1)1450 5290 y Fu(a)1501 5305 y Fp(1)1541 5290 y Fw(])22 b(+)g Fu(c)1730 5305 y Fp(2)1770 5290 y Fw([)p Fu(b)1838 5305 y Fp(2)1878 5290 y Fu(a)1929 5305 y Fp(2)1968 5290 y Fw(])h(+)2181 5166 y Fp(2)2127 5196 y Fm(X)2116 5406 y Fq(i;j)t Fp(=1)2282 5290 y Fw([)p Fu(b)2350 5305 y Fq(i)2378 5290 y Fu(a)2429 5305 y Fq(j)2466 5290 y Fw(])p Fu(a)2544 5249 y Fq(?)2544 5315 y(j)2584 5290 y Fu(b)2625 5249 y Fq(?)2625 5315 y(i)2665 5290 y Fu(:)94 5580 y Fw(T)-8 b(o)26 b(obtain)e(the)i(splitting)d(\()1104 5580 y SDict begin H.S end 1104 5580 a Fw(4.5)1229 5516 y SDict begin H.R end 1229 5516 a 1229 5580 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.4.5) cvn H.B /ANN pdfmark end 1229 5580 a Fw(\))i(of)g(\()1456 5555 y Fm(e)1434 5580 y Fu(A;)1564 5555 y Fm(e)1551 5580 y Fu(S)5 b Fw(\),)27 b(w)m(e)f(apply)f(the)h(automorphism)d Fu(')i Fw(of)g Fu(R)q Fr(hh)3272 5555 y Fm(e)3250 5580 y Fu(A)p Fr(ii)g Fw(\014xing)94 5696 y(all)f(arro)m(ws)i(except)h Fu(c)869 5711 y Fp(1)933 5696 y Fw(and)f Fu(c)1158 5711 y Fp(2)1197 5696 y Fw(,)h(and)f(suc)m(h)h(that)e Fu(')p Fw(\()p Fu(c)1995 5711 y Fq(i)2023 5696 y Fw(\))i(=)h Fu(c)2234 5711 y Fq(i)2269 5696 y Fr(\000)7 b Fu(a)2404 5660 y Fq(?)2404 5721 y(i)2445 5696 y Fu(b)2486 5660 y Fq(?)2486 5721 y(i)2526 5696 y Fw(.)41 b(An)26 b(easy)g(c)m(hec)m(k)i(sho)m(ws)f(that) p eop end end %%Page: 34 34 TeXDict begin HPSdict begin 34 33 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.34) cvn H.B /DEST pdfmark end 94 218 a Fs(34)492 b(HARM)25 b(DERKSEN,)f(JERZY)i(WEYMAN,)f(AND)f(ANDREI)g(ZELEVINSKY)94 422 y 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H.V] /Dest (page.35) cvn H.B /DEST pdfmark end 94 218 a 1179 w Fs(QUIVERS)31 b(WITH)i(POTENTIALS)g(I)1102 b(35)94 417 y Fw(the)37 b(cyclic)e(part)h(of)f Fu(R)q Fr(hh)p Fu(A)p Fr(ii)g Fw(is)h(the)g(closure)g(of)f(the)h(span)h(of)e(the)h (elemen)m(ts)h(\()p Fu(cba)p Fw(\))3222 381 y Fq(m)3289 417 y Fu(e)3334 432 y Fq(j)3406 417 y Fw(for)e(all)94 533 y(v)m(ertices)i Fu(j)42 b Fw(and)36 b(all)d Fu(m)h Fr(\025)f Fw(0.)53 b(Denoting)34 b(b)m(y)j Fu(p)e Fw(the)h(pro)5 b(jection)36 b Fu(R)q Fr(hh)p Fu(A)p Fr(ii)c(!)h Fw(T)-8 b(r\()p Fr(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\)\),)36 b(w)m(e)94 649 y(see)f(that)e(the)h(rigidit)m(y)e(of)h(\()p Fu(A;)17 b(S)6 b Fw(\))33 b(follo)m(ws)f(from)g(the)i(fact)g(that)f Fu(p)p Fw(\()p Fu(cbae)2774 664 y Fq(j)2811 649 y Fw(\))d(=)f(0)k(for)g (all)e Fu(j)6 b Fw(.)47 b(No)m(w)94 766 y(if)32 b Fu(ae)280 781 y Fq(j)345 766 y Fr(6)p Fw(=)27 b(0)32 b(and)h Fu(h)p Fw(\()p Fu(ae)909 781 y Fq(j)946 766 y Fw(\))28 b(=)f Fu(i)33 b Fw(then)g(w)m(e)h(ha)m(v)m(e)998 942 y Fu(p)p Fw(\()p Fu(cbae)1264 957 y Fq(j)1301 942 y Fw(\))28 b(=)f Fu(p)p Fw(\()p Fu(acbe)1736 957 y Fq(i)1765 942 y Fw(\))h(=)f Fu(p)p Fw(\()p Fu(cabe)2200 957 y Fq(i)2229 942 y Fw(\))h(=)f Fu(p)p Fw(\()p Fu(cbae)2664 957 y Fq(i)2693 942 y Fw(\))p Fu(:)94 1118 y Fw(Con)m(tin)m(uing)33 b(in)g(the)h(same)f(w)m(a)m(y)-8 b(,)35 b(w)m(e)g(obtain)d(that,)i(for)f(an)m(y)h Fu(m)c Fr(\025)f Fw(1)34 b(suc)m(h)h(that)e Fu(a)3158 1082 y Fq(m)3225 1118 y Fu(e)3270 1133 y Fq(j)3336 1118 y Fr(6)p Fw(=)c(0,)k(w)m(e)94 1234 y(ha)m(v)m(e)j Fu(p)p Fw(\()p Fu(cbae)587 1249 y Fq(j)625 1234 y Fw(\))30 b(=)h Fu(p)p Fw(\()p Fu(cbae)1066 1249 y Fq(k)1109 1234 y Fw(\),)k(where)h Fu(k)h Fw(is)d(the)h(end-p)s(oin)m(t)f(of)g(the)g(path)h Fu(a)2855 1198 y Fq(m)2922 1234 y Fu(e)2967 1249 y Fq(j)3003 1234 y Fw(.)49 b(T)-8 b(aking)34 b Fu(m)h Fw(the)94 1351 y(largest)d(suc)m(h)i(that)f Fu(a)892 1314 y Fq(m)959 1351 y Fu(e)1004 1366 y Fq(j)1068 1351 y Fr(6)p Fw(=)27 b(0,)33 b(w)m(e)g(conclude)h(that)e Fu(p)p Fw(\()p Fu(cbae)2302 1366 y Fq(j)2339 1351 y Fw(\))c(=)f(0,)33 b(as)f(desired.)194 1536 y(As)d(sho)m(wn)h(in)e([)763 1536 y SDict begin H.S end 763 1536 a Fw(27)861 1472 y SDict begin H.R end 861 1472 a 861 1536 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.kr) cvn H.B /ANN pdfmark end 861 1536 a Fw(],)h(the)g(quiv)m(er)h Fu(Q)p Fw(\(3\))e(in)g(Example)2141 1536 y SDict begin H.S end 2141 1536 a Fw(8.7)2266 1472 y SDict begin H.R end 2266 1472 a 2266 1536 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.8.7) cvn H.B /ANN pdfmark end 2266 1536 a 29 w Fw(is)g Fv(not)38 b Fw(m)m(utation-equiv)-5 b(alen)m(t)26 b(to)i(an)94 1652 y(acyclic)34 b(one.)48 b(So)33 b(there)i(exist)f(QPs) h(with)e(\014nite-dimensional)e(Jacobian)i(algebras)g(\(and)h(also)94 1768 y(rigid)d(QPs\),)i(whic)m(h)g(are)g(not)f(m)m(utation-equiv)-5 b(alen)m(t)31 b(to)h(acyclic)g(ones.)194 1885 y(W)-8 b(e)30 b(no)m(w)g(describ)s(e)g(a)f(pro)s(cedure)h(to)f(obtain)g(new)h (QPs)g(with)f(\014nite-dimensional)d(Jacobian)94 2001 y(algebras)32 b(\(and)h(new)h(rigid)c(QPs\))j(from)f(old)f(ones.)94 2026 y SDict begin H.S end 94 2026 a 94 2026 a SDict begin 14 H.A end 94 2026 a 94 2026 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.8.8) cvn H.B /DEST pdfmark end 94 2026 a 160 x FA(De\014nition)36 b(8.8.)41 b Fw(F)-8 b(or)31 b(a)h(QP)g(\()p Fu(A;)17 b(S)6 b Fw(\))31 b(and)h(a)g(subset)h Fu(I)40 b Fw(of)31 b(the)i(v)m(ertex)g(set)g Fu(Q)3003 2201 y Fp(0)3043 2186 y Fw(,)f(w)m(e)h(de\014ne)g(the)94 2302 y Fv(r)-5 b(estriction)40 b Fw(of)32 b(\()p Fu(A;)17 b(S)6 b Fw(\))32 b(to)h Fu(I)40 b Fw(as)33 b(the)g(QP)g(\()p Fu(A)p Fr(j)1777 2317 y Fq(I)1816 2302 y Fu(;)17 b(S)6 b Fr(j)1954 2317 y Fq(I)1993 2302 y Fw(\))33 b(giv)m(en)g(b)m(y)1578 2500 y Fu(A)p Fr(j)1679 2515 y Fq(I)1746 2500 y Fw(=)1854 2406 y Fm(M)1850 2617 y Fq(i;j)t Fn(2)p Fq(I)2025 2500 y Fu(A)2098 2515 y Fq(i;j)94 2777 y Fw(and)1610 2913 y Fu(S)6 b Fr(j)1704 2928 y Fq(I)1771 2913 y Fw(=)27 b Fu( )1937 2928 y Fq(I)1978 2913 y Fw(\()p Fu(S)6 b Fw(\))p Fu(;)94 3069 y Fw(where)25 b Fu( )430 3084 y Fq(I)498 3069 y Fw(:)j Fu(R)q Fr(hh)p Fu(A)p Fr(ii)f(!)g Fu(R)q Fr(hh)p Fu(A)p Fr(j)1265 3084 y Fq(I)1305 3069 y Fr(ii)c Fw(is)g(the)h(homomorphism)c(of)k Fu(R)q Fw(-algebras)e(suc)m (h)j(that)e Fu( )3390 3084 y Fq(I)3431 3069 y Fw(\()p Fu(a)p Fw(\))28 b(=)94 3185 y Fu(a)33 b Fw(for)f Fu(a)c Fr(2)g Fu(A)p Fr(j)601 3200 y Fq(I)641 3185 y Fw(,)33 b(and)f Fu( )953 3200 y Fq(I)994 3185 y Fw(\()p Fu(b)p Fw(\))c(=)f(0)32 b(for)g(an)m(y)i(arro)m(w)e Fu(b)h Fw(not)g(b)s (elonging)e(to)h Fu(A)p Fr(j)2840 3200 y Fq(I)2879 3185 y Fw(.)94 3279 y SDict begin H.S end 94 3279 a 94 3279 a SDict begin 14 H.A end 94 3279 a 94 3279 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.8.9) cvn H.B /DEST pdfmark end 94 3279 a 91 x FA(Prop)s(osition)d(8.9.)37 b Fv(The)28 b(homomorphism)f Fu( )1838 3385 y Fq(I)1907 3370 y Fv(induc)-5 b(es)29 b(an)f(epimorphism)f(of)i(Jac)-5 b(obian)28 b(alge-)94 3487 y(br)-5 b(as)30 b Fr(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\))28 b Fr(!)f(P)8 b Fw(\()p Fu(A)p Fr(j)1002 3502 y Fq(I)1042 3487 y Fu(;)17 b(S)6 b Fr(j)1180 3502 y Fq(I)1219 3487 y Fw(\))30 b Fv(and)g(an)f(epimorphism)f(of)i(deformation)f(sp)-5 b(ac)g(es)29 b Fw(Def)7 b(\()p Fu(A;)17 b(S)6 b Fw(\))27 b Fr(!)94 3603 y Fw(Def)7 b(\()p Fu(A)p Fr(j)388 3618 y Fq(I)428 3603 y Fu(;)17 b(S)6 b Fr(j)566 3618 y Fq(I)605 3603 y Fw(\))p Fv(.)48 b(Ther)-5 b(efor)g(e,)36 b(if)g Fr(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\))36 b Fv(is)g (\014nite-dimensional,)e(or)i(if)g Fw(\()p Fu(A;)17 b(S)6 b Fw(\))36 b Fv(is)g(rigid,)g(then)94 3719 y(the)f(same)f(is)h(true)h (for)e Fw(\()p Fu(A)p Fr(j)1110 3734 y Fq(I)1150 3719 y Fu(;)17 b(S)6 b Fr(j)1288 3734 y Fq(I)1327 3719 y Fw(\))p Fv(.)94 3904 y(Pr)-5 b(o)g(of.)42 b Fw(Remem)m(b)s(ering)36 b(\()1055 3904 y SDict begin H.S end 1055 3904 a Fw(3.1)1180 3840 y SDict begin H.R end 1180 3840 a 1180 3904 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.3.1) cvn H.B /ANN pdfmark end 1180 3904 a Fw(\),)j(it)d(is)i(easy)h(to)e(see)i (that)f Fu( )2277 3919 y Fq(I)2317 3904 y Fw(\()p Fu(@)2406 3919 y Fq(a)2448 3904 y Fu(S)6 b Fw(\))36 b(=)h Fu(@)2752 3919 y Fq(a)2794 3904 y Fu( )2857 3919 y Fq(I)2897 3904 y Fw(\()p Fu(S)6 b Fw(\))38 b(for)f(an)m(y)h(arro)m(w)94 4021 y Fu(a)31 b Fr(2)f Fu(A)p Fr(j)373 4036 y Fq(I)413 4021 y Fw(,)k(and)g Fu( )728 4036 y Fq(I)768 4021 y Fw(\()p Fu(@)857 4036 y Fq(b)892 4021 y Fu(S)6 b Fw(\))30 b(=)f(0)34 b(for)f(an)m(y)i(arro)m(w)f Fu(b)g Fw(not)g(b)s(elonging)e(to)i Fu(A)p Fr(j)2740 4036 y Fq(I)2779 4021 y Fw(.)48 b(Therefore,)35 b(w)m(e)g(ha)m(v)m(e)94 4137 y Fu( )157 4152 y Fq(I)198 4137 y Fw(\()p Fu(J)9 b Fw(\()p Fu(S)d Fw(\)\))27 b(=)h Fu(J)9 b Fw(\()p Fu( )774 4152 y Fq(I)814 4137 y Fw(\()p Fu(S)d Fw(\)\),)32 b(implying)e(all)h(the)i(assertions.)1383 b Fh(\003)94 4235 y SDict begin H.S end 94 4235 a 94 4235 a SDict begin 14 H.A end 94 4235 a 94 4235 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.8.10) cvn H.B /DEST pdfmark end 94 4235 a 87 x FA(Corollary)30 b(8.10.)37 b Fv(Supp)-5 b(ose)28 b(that)i Fu(A)e Fv(and)h Fu(A)1770 4286 y Fn(0)1822 4322 y Fv(ar)-5 b(e)29 b Fw(2)p Fv(-acyclic,)g(and)f(ther)-5 b(e)29 b(is)f(a)h(rigid)f(QP)h Fw(\()p Fu(A;)17 b(S)6 b Fw(\))94 4438 y Fv(on)42 b Fu(A)p Fv(.)66 b(L)-5 b(et)43 b Fu(B)5 b Fw(\()p Fu(A)p Fw(\))42 b Fv(and)f Fu(B)5 b Fw(\()p Fu(A)1242 4402 y Fn(0)1266 4438 y Fw(\))42 b Fv(b)-5 b(e)41 b(the)h(c)-5 b(orr)g(esp)g(onding)41 b(skew-symmetric)f(inte)-5 b(ger)42 b(matric)-5 b(es)94 4555 y(given)39 b(by)h Fw(\()523 4555 y SDict begin H.S end 523 4555 a Fw(7.1)647 4490 y SDict begin H.R end 647 4490 a 647 4555 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.7.1) cvn H.B /ANN pdfmark end 647 4555 a Fw(\))p Fv(.)59 b(Supp)-5 b(ose)39 b(that)h Fu(B)5 b Fw(\()p Fu(A)1544 4518 y Fn(0)1567 4555 y Fw(\))40 b Fv(c)-5 b(an)39 b(b)-5 b(e)39 b(obtaine)-5 b(d)39 b(by)g(a)h(simultane)-5 b(ous)39 b(p)-5 b(ermutation)94 4671 y(of)33 b(r)-5 b(ows)33 b(and)g(c)-5 b(olumns)32 b(fr)-5 b(om)33 b(some)f(princip)-5 b(al)32 b(submatrix)h(of)g(a)g(matrix)g(mutation-e)-5 b(quivalent)94 4787 y(to)36 b Fu(B)5 b Fw(\()p Fu(A)p Fw(\))p Fv(.)44 b(Then)34 b(ther)-5 b(e)35 b(exists)g(a)f(rigid)h(QP)g Fw(\()p Fu(A)1880 4751 y Fn(0)1903 4787 y Fu(;)17 b(S)2013 4751 y Fn(0)2036 4787 y Fw(\))35 b Fv(on)f Fu(A)2321 4751 y Fn(0)2344 4787 y Fv(.)94 4972 y(Pr)-5 b(o)g(of.)42 b Fw(In)f(view)h(of)f(\()915 4972 y SDict begin H.S end 915 4972 a Fw(7.1)1039 4908 y SDict begin H.R end 1039 4908 a 1039 4972 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.7.1) cvn H.B /ANN pdfmark end 1039 4972 a Fw(\),)j(the)d(matrix)f Fu(B)5 b Fw(\()p Fu(A)p Fr(j)1868 4987 y Fq(I)1908 4972 y Fw(\))41 b(is)g(the)h(principal)d(submatrix)h(of)h Fu(B)5 b Fw(\()p Fu(A)p Fw(\))42 b(in-)94 5089 y(v)m(olving)h(ro)m(ws)h (and)f(columns)f(from)g Fu(I)8 b Fw(.)75 b(Therefore,)47 b(our)c(assertion)g(follo)m(ws)f(b)m(y)i(com)m(bining)94 5205 y(Prop)s(osition)619 5205 y SDict begin H.S end 619 5205 a Fw(8.9)744 5141 y SDict begin H.R end 744 5141 a 744 5205 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.8.9) cvn H.B /ANN pdfmark end 744 5205 a 33 w Fw(with)32 b(Corollary)1430 5205 y SDict begin H.S end 1430 5205 a Fw(6.11)1603 5141 y SDict begin H.R end 1603 5141 a 1603 5205 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.6.11) cvn H.B /ANN pdfmark end 1603 5205 a 32 w Fw(and)h(Prop)s(osition)2350 5205 y SDict begin H.S end 2350 5205 a Fw(7.1)2475 5141 y SDict begin H.R end 2475 5141 a 2475 5205 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.7.1) cvn H.B /ANN pdfmark end 2475 5205 a Fw(.)1082 b Fh(\003)194 5395 y Fw(W)-8 b(e)33 b(conclude)g(this)f(section)h(with)f(a)h(com)m(binatorial)c (application)h(of)i(Corollary)3244 5395 y SDict begin H.S end 3244 5395 a Fw(8.10)3417 5330 y SDict begin H.R end 3417 5330 a 3417 5395 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.8.10) cvn H.B /ANN pdfmark end 3417 5395 a Fw(.)94 5414 y SDict begin H.S end 94 5414 a 94 5414 a SDict begin 14 H.A end 94 5414 a 94 5414 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.8.11) cvn H.B /DEST pdfmark end 94 5414 a 166 x FA(Corollary)j(8.11.)40 b Fv(L)-5 b(et)33 b Fu(B)g Fw(=)27 b Fu(B)5 b Fw(\()p Fu(A)p Fw(\()p Fu(n)p Fw(\)\))33 b Fv(b)-5 b(e)33 b(the)f(matrix)h(asso)-5 b(ciate)g(d)31 b(with)i(the)f(triangular)h(grid)94 5696 y(of)j(some)g(or)-5 b(der)36 b Fu(n)g Fv(\(se)-5 b(e)36 b(Example)1410 5716 y SDict begin H.S end 1410 5716 a -20 x Fv(8.7)1540 5632 y SDict begin H.R end 1540 5632 a 1540 5696 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.8.7) cvn H.B /ANN pdfmark end 1540 5696 a Fv(\).)48 b(Then)36 b(none)f(of)h(the)g(matric)-5 b(es)36 b(mutation-e)-5 b(quivalent)p eop end end %%Page: 36 36 TeXDict begin HPSdict begin 36 35 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.36) cvn H.B /DEST pdfmark end 94 218 a Fs(36)492 b(HARM)25 b(DERKSEN,)f(JERZY)i(WEYMAN,)f(AND)f(ANDREI)g(ZELEVINSKY)94 417 y Fv(to)36 b Fu(B)k Fv(c)-5 b(ontains)1402 649 y Fu(B)1481 608 y Fn(0)1532 649 y Fw(=)1635 449 y Fm(0)1635 628 y(@)1761 531 y Fw(0)160 b(2)122 b Fr(\000)p Fw(2)1722 648 y Fr(\000)p Fw(2)g(0)161 b(2)1761 764 y(2)122 b Fr(\000)p Fw(2)g(0)2267 449 y Fm(1)2267 628 y(A)94 921 y Fv(as)35 b(a)g(princip)-5 b(al)33 b(submatrix.)94 1107 y(Pr)-5 b(o)g(of.)42 b Fw(Note)33 b(that)h Fu(B)921 1071 y Fn(0)973 1107 y Fw(=)c Fu(B)5 b Fw(\()p Fu(A)1269 1071 y Fn(0)1292 1107 y Fw(\),)34 b(where)h Fu(A)1747 1071 y Fn(0)1804 1107 y Fw(is)e(the)h(quiv)m(er)g(in)f(Example)2889 1107 y SDict begin H.S end 2889 1107 a Fw(8.6)3014 1043 y SDict begin H.R end 3014 1043 a 3014 1107 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.8.6) cvn H.B /ANN pdfmark end 3014 1107 a Fw(.)46 b(W)-8 b(e)34 b(ha)m(v)m(e)h(seen)94 1223 y(that)43 b(that)g(there)g(exists)h(a)f(rigid)e(QP)i(on)g Fu(A)p Fw(,)i(but)e(not)g(on)g Fu(A)2475 1187 y Fn(0)2498 1223 y Fw(.)75 b(Th)m(us)44 b(our)f(statemen)m(t)g(is)g(a)94 1340 y(sp)s(ecial)32 b(case)i(of)e(Corollary)1160 1340 y SDict begin H.S end 1160 1340 a Fw(8.10)1334 1275 y SDict begin H.R end 1334 1275 a 1334 1340 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.8.10) cvn H.B /ANN pdfmark end 1334 1340 a Fw(.)2223 b Fh(\003)94 1433 y SDict begin H.S end 94 1433 a 94 1433 a SDict begin 14 H.A end 94 1433 a 94 1433 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.8.12) cvn H.B /DEST pdfmark end 94 1433 a 92 x FA(Remark)47 b(8.12.)f Fw(Using)40 b(the)h(results)g(in)f([)1730 1525 y SDict begin H.S end 1730 1525 a Fw(1)1778 1461 y SDict begin H.R end 1778 1461 a 1778 1525 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.ca3) cvn H.B /ANN pdfmark end 1778 1525 a Fw(,)j(Section)d(2.6])h(\(cf.)67 b(also)40 b([)2826 1525 y SDict begin H.S end 2826 1525 a Fw(22)2923 1461 y SDict begin H.R end 2923 1461 a 2923 1525 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.gls) cvn H.B /ANN pdfmark end 2923 1525 a Fw(,)j(Theorem)e(1]\),)h(it)94 1641 y(is)35 b(easy)i(to)e(see)h(that)g(the)f(quiv)m(er)i Fu(Q)p Fw(\()p Fu(n)p Fw(\))e(in)g(Example)2150 1641 y SDict begin H.S end 2150 1641 a Fw(8.7)2275 1577 y SDict begin H.R end 2275 1577 a 2275 1641 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.8.7) cvn H.B /ANN pdfmark end 2275 1641 a 35 w Fw(is)g(naturally)f(asso)s(ciated)h(with)g(the)94 1758 y(cluster)e(algebra)d(structure)k(in)d(the)h(co)s(ordinate)f(ring) g(of)g(the)i(base)f(a\016ne)h(space)g(of)e(the)i(group)94 1874 y Fu(S)6 b(L)226 1889 y Fq(n)p Fp(+3)364 1874 y Fw(.)41 b(W)-8 b(e)26 b(ha)m(v)m(e)i(b)s(een)e(informed)f(b)m(y)i(J.)f (Sc)m(hr\177)-49 b(oer)27 b(that,)g(follo)m(wing)c(his)j(suggestion,)h (A.)g(Sev)m(en)94 1990 y(has)d(sho)m(wn)g(that)f(a)f(sk)m(ew-symmetric) i(matrix)d Fu(B)1903 1954 y Fn(0)1950 1990 y Fw(asso)s(ciated)h(with)h (an)g(arbitrary)f(tree)h(app)s(ears)94 2106 y(as)31 b(a)e(principal)f (submatrix)h(in)g(some)h(matrix)e(m)m(utation-equiv)-5 b(alen)m(t)28 b(to)i(the)g(matrix)e Fu(B)5 b Fw(\()p Fu(A)p Fw(\()p Fu(n)p Fw(\)\))94 2223 y(for)32 b(some)f Fu(n)p Fw(.)44 b(J.)31 b(Sc)m(hr\177)-49 b(oer)33 b(also)e(informed)f (us)i(that)g(Corollary)2440 2223 y SDict begin H.S end 2440 2223 a Fw(8.11)2613 2158 y SDict begin H.R end 2613 2158 a 2613 2223 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.8.11) cvn H.B /ANN pdfmark end 2613 2223 a 32 w Fw(has)g(b)s(een)h (also)d(pro)m(v)m(ed)j(b)m(y)94 2339 y(B.)g(Keller)e(\(using)i(a)f (di\013eren)m(t)h(metho)s(d\).)94 2502 y SDict begin H.S end 94 2502 a 94 2502 a SDict begin 14 H.A end 94 2502 a 94 2502 a SDict begin [ /View [/XYZ H.V] /Dest (section.9) cvn H.B /DEST pdfmark end 94 2502 a 873 2593 a Fw(9.)49 b Fx(Rela)-7 b(tion)38 b(to)g(cluster-til)-7 b(ted)36 b(algebras)194 2767 y Fw(Let)j Fu(Q)g Fw(b)s(e)g(a)f(quiv)m(er)h(with)f (the)i(v)m(ertex)g(span)f Fu(R)h Fw(and)e(the)h(arro)m(w)g(span)g Fu(A)p Fw(.)62 b(Assume)40 b(that)94 2884 y Fu(Q)32 b Fw(is)f(2-acyclic.)42 b(Let)32 b Fu(B)5 b Fw(\()p Fu(A)p Fw(\))32 b(b)s(e)g(the)g(corresp)s(onding)f(sk)m(ew-symmetric)i(in)m (teger)e(matrix)f(giv)m(en)94 3000 y(b)m(y)37 b(\()271 3000 y SDict begin H.S end 271 3000 a Fw(7.1)395 2936 y SDict begin H.R end 395 2936 a 395 3000 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.7.1) cvn H.B /ANN pdfmark end 395 3000 a Fw(\).)51 b(As)36 b(sho)m(wn)g(in)f([)1100 3000 y SDict begin H.S end 1100 3000 a Fw(19)1197 2936 y SDict begin H.R end 1197 2936 a 1197 3000 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.ca2) cvn H.B /ANN pdfmark end 1197 3000 a Fw(],)h(the)g(matrix)d Fu(B)5 b Fw(\()p Fu(A)p Fw(\))36 b(giv)m(es)f(rise)g(to)g(a)g(cluster)g(algebra)f(of)h (\014nite)94 3116 y(t)m(yp)s(e)30 b(if)d(and)i(only)f(if)f Fu(Q)i Fw(is)f(m)m(utation-equiv)-5 b(alen)m(t)26 b(to)j(a)f(Dynkin)g (quiv)m(er)i(\(that)e(is,)h(an)f(arbitrary)94 3232 y(orien)m(tation)41 b(of)h(a)g(Dynkin)h(diagram)d(of)i(one)h(of)f(the)h(t)m(yp)s(es)h Fu(A)2507 3247 y Fq(n)2554 3232 y Fw(,)h Fu(D)2707 3247 y Fq(n)2754 3232 y Fw(,)g Fu(E)2898 3247 y Fp(6)2937 3232 y Fw(,)g Fu(E)3081 3247 y Fp(7)3121 3232 y Fw(,)g(or)d Fu(E)3394 3247 y Fp(8)3434 3232 y Fw(\).)73 b(In)94 3349 y(particular,)33 b(as)h(w)m(e)g(ha)m(v)m(e)h(seen)g(in)e(Example)1798 3349 y SDict begin H.S end 1798 3349 a Fw(8.3)1923 3284 y SDict begin H.R end 1923 3284 a 1923 3349 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.8.3) cvn H.B /ANN pdfmark end 1923 3349 a Fw(,)h(there)g(is)g(a)f(rigid)f(reduced)j(QP)f(\()p Fu(A;)17 b(S)6 b Fw(\),)33 b(and)94 3465 y(it)f(is)g(unique)i(up)f(to)f (a)h(righ)m(t-equiv)-5 b(alence;)31 b(in)h(fact,)h(\(the)g(righ)m (t-equiv)-5 b(alence)32 b(class)h(of)7 b(\))32 b(\()p Fu(A;)17 b(S)6 b Fw(\))94 3581 y(is)31 b(obtained)f(b)m(y)i(a)e (sequence)k(of)c(m)m(utations)g(from)g(a)g(QP)h(\()p Fu(A)2364 3596 y Fp(0)2404 3581 y Fu(;)17 b Fw(0\),)30 b(where)i Fu(A)2945 3596 y Fp(0)3016 3581 y Fw(is)e(asso)s(ciated)h(to) 94 3697 y(a)39 b(Dynkin)g(quiv)m(er.)63 b(W)-8 b(e)39 b(no)m(w)h(presen)m(t)g(an)f(explicit)f(c)m(hoice)h(of)f(suc)m(h)j(a)d (p)s(oten)m(tial)f Fu(S)6 b Fw(.)63 b(T)-8 b(o)38 b(do)94 3814 y(this,)33 b(w)m(e)g(need)h(some)e(preparation.)194 3930 y(First)g(note)h(that,)h(according)e(to)h([)1475 3930 y SDict begin H.S end 1475 3930 a Fw(19)1572 3866 y SDict begin H.R end 1572 3866 a 1572 3930 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.ca2) cvn H.B /ANN pdfmark end 1572 3930 a Fw(,)h(Theorem)f(1.8],)g(ev)m(ery)i(quiv)m(er)f(m)m (utation-equiv)-5 b(alen)m(t)94 4046 y(to)26 b(to)g(a)g(Dynkin)g(quiv)m (er)h(has)g(no)f(m)m(ultiple)d(edges,)29 b(that)d(is)g Fr(j)p Fu(b)2344 4061 y Fq(i;j)2424 4046 y Fr(j)h(\024)i Fw(1)c(for)h(ev)m(ery)i(en)m(try)f(of)f Fu(B)5 b Fw(\()p Fu(A)p Fw(\).)94 4162 y(In)33 b(other)g(w)m(ords,)h(w)m(e)f(ha)m(v)m (e:)94 4339 y(\(9.1\))1298 4339 y SDict begin H.S end 1298 4339 a 1298 4339 a SDict begin 14 H.A end 1298 4339 a 1298 4339 a SDict begin [ /View [/XYZ H.V] /Dest (equation.9.1) cvn H.B /DEST pdfmark end 1298 4339 a Fw(dim)15 b Fu(A)1550 4354 y Fq(i;j)1658 4339 y Fr(\024)28 b Fw(1)33 b(for)f(all)e Fu(i)j Fw(and)g Fu(j)6 b Fw(.)94 4516 y(Therefore,)26 b(w)m(e)d(can)g(unam)m(biguously)e(denote)i(b)m(y)g Fu(a)1999 4531 y Fq(i;j)2102 4516 y Fw(the)g(only)e(arro)m(w)i(in)e(a)h(non-zero) g(subspace)94 4632 y Fu(A)167 4647 y Fq(i;j)248 4632 y Fw(.)43 b(W)-8 b(e)33 b(will)d(also)i(use)i(the)f(con)m(v)m(en)m (tion)g(that)g Fu(a)1953 4647 y Fq(i;j)2061 4632 y Fw(=)27 b(0)33 b(whenev)m(er)i Fu(A)2747 4647 y Fq(i;j)2855 4632 y Fw(=)27 b Fr(f)p Fw(0)p Fr(g)p Fw(.)194 4748 y(Second,)46 b(w)m(e)e(use)f(the)g(follo)m(wing)d(terminology:)61 b(a)42 b Fv(chor)-5 b(d)5 b(less)43 b(cycle)49 b Fw(in)42 b(\(the)h(underlying)94 4864 y(graph)37 b(of)7 b(\))36 b Fu(Q)h Fw(is)f(a)g Fu(d)p Fw(-subset)h(of)f(v)m(ertices)i Fu(I)k Fr(\022)35 b Fu(Q)1968 4879 y Fp(0)2044 4864 y Fw(that)i(can)f(b)s(e)h(bijectiv)m(ely)f(lab)s(eled)f(b)m(y)j(the)94 4981 y(elemen)m(ts)26 b(of)e Fd(Z)p Fu(=d)p Fd(Z)19 b Fw(so)26 b(that)e(the)i(edges)g(b)s(et)m(w)m(een)h(them)e(are)g (precisely)g Fr(f)p Fu(i;)17 b(i)7 b Fw(+)g(1)p Fr(g)24 b Fw(for)g Fu(i)k Fr(2)g Fd(Z)p Fu(=d)p Fd(Z)p Fw(.)94 5097 y(According)39 b(to)f([)716 5097 y SDict begin H.S end 716 5097 a Fw(19)814 5033 y SDict begin H.R end 814 5033 a 814 5097 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.ca2) cvn H.B /ANN pdfmark end 814 5097 a Fw(,)i(Prop)s(osition)c(9.7],)k (if)e Fu(Q)g Fw(is)g(m)m(utation-equiv)-5 b(alen)m(t)37 b(to)h(a)g(Dynkin)h(quiv)m(er)94 5213 y(then)d(the)g(arro)m(ws)g(of)f (ev)m(ery)i(c)m(hordless)f(cycle)g(in)e Fu(Q)i Fw(m)m(ust)f(b)s(e)h (cyclically)d(orien)m(ted.)52 b(In)35 b(terms)94 5329 y(of)e(the)g(arro)m(w)f(span)h Fu(A)p Fw(,)g(this)f(condition)g(can)g (b)s(e)h(stated)g(as)g(follo)m(ws:)477 5506 y(F)-8 b(or)32 b(an)m(y)h(c)m(hordless)g Fu(d)p Fw(-cycle)g Fu(I)8 b Fw(,)32 b(there)i(exists)f(a)f(bijection)g Fu(\027)i Fw(:)27 b Fd(Z)p Fu(=d)p Fd(Z)22 b Fr(!)28 b Fu(I)40 b Fw(suc)m(h)94 5506 y SDict begin H.S end 94 5506 a 94 5506 a SDict begin 17.5 H.A end 94 5506 a 94 5506 a SDict begin [ /View [/XYZ H.V] /Dest (equation.9.2) cvn H.B /DEST pdfmark end 94 5506 a Fw(\(9.2\))477 5657 y(that)32 b(the)h(set)g(of)g (arro)m(ws)g(in)e(the)i(restriction)f Fu(A)p Fr(j)2281 5672 y Fq(I)2353 5657 y Fw(is)g Fr(f)p Fu(a)2552 5673 y Fq(\027)t Fp(\()p Fq(i)p Fp(\))p Fq(;\027)t Fp(\()p Fq(i)p Fp(+1\))2930 5657 y Fr(j)27 b Fu(i)h Fr(2)g Fd(Z)p Fu(=d)p Fd(Z)p Fr(g)-6 b Fu(:)p eop end end %%Page: 37 37 TeXDict begin HPSdict begin 37 36 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.37) cvn H.B /DEST pdfmark end 94 218 a 1179 w Fs(QUIVERS)31 b(WITH)i(POTENTIALS)g(I)1102 b(37)94 417 y Fw(\(see)36 b(De\014nition)747 417 y SDict begin H.S end 747 417 a Fw(8.8)872 353 y SDict begin H.R end 872 353 a 872 417 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.8.8) cvn H.B /ANN pdfmark end 872 417 a Fw(\).)48 b(Note)34 b(that)g(the)h(c)m(hoice)g(of)e(a)h(bijection)f Fu(\027)41 b Fw(in)33 b(\()2734 417 y SDict begin H.S end 2734 417 a Fw(9.2)2859 353 y SDict begin H.R end 2859 353 a 2859 417 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.9.2) cvn H.B /ANN pdfmark end 2859 417 a Fw(\))h(is)g(unique)g(up)h(to)f (a)94 533 y(cyclic)f(shift.)194 649 y(W)-8 b(e)33 b(call)e(a)h(p)s (oten)m(tial)f Fu(S)38 b Fw(on)33 b Fu(A)g Fv(primitive)39 b Fw(if)31 b(it)h(has)h(the)g(form)94 842 y(\(9.3\))1017 842 y SDict begin H.S end 1017 842 a 1017 842 a SDict begin 14 H.A end 1017 842 a 1017 842 a SDict begin [ /View [/XYZ H.V] /Dest (equation.9.3) cvn H.B /DEST pdfmark end 1017 842 a Fu(S)g Fw(=)1214 747 y Fm(X)1268 958 y Fq(I)1374 842 y Fu(x)1429 857 y Fq(I)1469 842 y Fu(a)1520 857 y Fq(\027)t Fp(\(1\))p Fq(;\027)t Fp(\(2\))1819 842 y Fr(\001)17 b(\001)g(\001)e Fu(a)2003 857 y Fq(\027)t Fp(\()p Fq(d)p Fn(\000)p Fp(1\))p Fq(;\027)t Fp(\()p Fq(d)p Fp(\))2378 842 y Fu(a)2429 857 y Fq(\027)t Fp(\()p Fq(d)p Fp(\))p Fq(;\027)t Fp(\(1\))2712 842 y Fu(;)94 1111 y Fw(where)50 b(the)e(\(\014nite\))g(sum)g(is)f(o)m(v)m(er)i(all)d(c)m(hordless)j (cycles)g Fu(I)56 b Fw(in)47 b Fu(Q)p Fw(,)52 b(for)c(eac)m(h)h Fu(I)55 b Fw(there)49 b(is)f(a)94 1228 y(bijection)33 b Fu(\027)40 b Fw(c)m(hosen)35 b(as)f(in)f(\()1169 1228 y SDict begin H.S end 1169 1228 a Fw(9.2)1294 1163 y SDict begin H.R end 1294 1163 a 1294 1228 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.9.2) cvn H.B /ANN pdfmark end 1294 1228 a Fw(\),)h(and)g(the)g(co)s(e\016cien)m(ts)h Fu(x)2303 1243 y Fq(I)2377 1228 y Fw(are)e(some)h(non-zero)g(elemen)m (ts)g(of)94 1344 y(the)f(base)h(\014eld)e Fu(K)7 b Fw(.)94 1344 y SDict begin H.S end 94 1344 a 94 1344 a SDict begin 14 H.A end 94 1344 a 94 1344 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.9.1) cvn H.B /DEST pdfmark end 94 1344 a 185 x FA(Prop)s(osition)39 b(9.1.)k Fv(If)36 b(a)h(quiver)g Fu(Q)h Fv(with)f(the)g(arr)-5 b(ow)36 b(sp)-5 b(an)37 b Fu(A)g Fv(is)g(mutation-e)-5 b(quivalent)36 b(to)i(a)94 1645 y(Dynkin)d(quiver,)f(and)g Fu(S)41 b Fv(is)35 b(a)f(primitive)h(p) -5 b(otential)34 b(on)h Fu(A)p Fv(,)f(then)h(the)g(QP)g Fw(\()p Fu(A;)17 b(S)6 b Fw(\))34 b Fv(is)h(rigid.)194 1831 y Fw(In)e(view)g(of)f(Corollary)1083 1831 y SDict begin H.S end 1083 1831 a Fw(6.11)1256 1766 y SDict begin H.R end 1256 1766 a 1256 1831 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.6.11) cvn H.B /ANN pdfmark end 1256 1831 a Fw(,)h(the)g(assertion)g(of)f(Prop)s(osition)2528 1831 y SDict begin H.S end 2528 1831 a Fw(9.1)2652 1766 y SDict begin H.R end 2652 1766 a 2652 1831 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.9.1) cvn H.B /ANN pdfmark end 2652 1831 a 33 w Fw(is)g(a)g(consequence)k(of)c(the)94 1947 y(follo)m(wing)e(lemma.)94 1966 y SDict begin H.S end 94 1966 a 94 1966 a SDict begin 14 H.A end 94 1966 a 94 1966 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.9.2) cvn H.B /DEST pdfmark end 94 1966 a 166 x FA(Lemma)50 b(9.2.)d Fv(Supp)-5 b(ose)44 b(the)h(arr)-5 b(ow)44 b(sp)-5 b(an)44 b Fu(A)h Fv(of)f(a)h(quiver)f Fu(Q)h Fv(satis\014es)f Fw(\()2973 2132 y SDict begin H.S end 2973 2132 a Fw(4.13)3147 2068 y SDict begin H.R end 3147 2068 a 3147 2132 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.4.13) cvn H.B /ANN pdfmark end 3147 2132 a Fw(\))p Fv(,)j Fw(\()3300 2132 y SDict begin H.S end 3300 2132 a Fw(9.1)3424 2068 y SDict begin H.R end 3424 2068 a 3424 2132 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.9.1) cvn H.B /ANN pdfmark end 3424 2132 a Fw(\))e Fv(and)94 2253 y Fw(\()132 2253 y SDict begin H.S end 132 2253 a Fw(9.2)257 2189 y SDict begin H.R end 257 2189 a 257 2253 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.9.2) cvn H.B /ANN pdfmark end 257 2253 a Fw(\))p Fv(,)34 b(and)g Fu(S)40 b Fv(is)34 b(a)f(primitive)h(p)-5 b(otential)34 b(on)f Fu(A)p Fv(.)45 b(L)-5 b(et)35 b Fw(\()p 2142 2173 74 4 v Fu(A;)p 2259 2173 66 4 v 17 w(S)5 b Fw(\))28 b(=)f Fu(\026)2552 2268 y Fq(k)2595 2253 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\))33 b Fv(for)h(some)g(vertex)f Fu(k)s Fv(,)94 2375 y(and)42 b(supp)-5 b(ose)p 654 2295 74 4 v 41 w Fu(A)42 b Fv(also)f(satis\014es) g Fw(\()1386 2375 y SDict begin H.S end 1386 2375 a Fw(4.13)1559 2310 y SDict begin H.R end 1559 2310 a 1559 2375 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.4.13) cvn H.B /ANN pdfmark end 1559 2375 a Fw(\))p Fv(,)i Fw(\()1708 2375 y SDict begin H.S end 1708 2375 a Fw(9.1)1833 2310 y SDict begin H.R end 1833 2310 a 1833 2375 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.9.1) cvn H.B /ANN pdfmark end 1833 2375 a Fw(\))f Fv(and)f Fw(\()2147 2375 y SDict begin H.S end 2147 2375 a Fw(9.2)2271 2310 y SDict begin H.R end 2271 2310 a 2271 2375 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.9.2) cvn H.B /ANN pdfmark end 2271 2375 a Fw(\))p Fv(.)66 b(Then)41 b(the)g(QP)h Fw(\()p 3055 2295 74 4 v Fu(A;)p 3172 2295 66 4 v 17 w(S)5 b Fw(\))42 b Fv(is)g(right-)94 2496 y(e)-5 b(quivalent)35 b(to)g(a)g(QP)f(on)p 1065 2416 74 4 v 35 w Fu(A)h Fv(with)f(a)h (primitive)f(p)-5 b(otential.)94 2690 y(Pr)g(o)g(of.)42 b Fw(Let)e Fm(e)-60 b Fu(\026)629 2705 y Fq(k)672 2690 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\))32 b(=)h(\()1133 2665 y Fm(e)1110 2690 y Fu(A;)1240 2665 y Fm(e)1227 2690 y Fu(S)6 b Fw(\))35 b(b)s(e)h(the)g(QP)g(giv)m(en)g(b)m(y)g(\()2285 2690 y SDict begin H.S end 2285 2690 a Fw(5.6)2410 2626 y SDict begin H.R end 2410 2626 a 2410 2690 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.6) cvn H.B /ANN pdfmark end 2410 2690 a Fw(\),)h(\()2550 2690 y SDict begin H.S end 2550 2690 a Fw(5.7)2674 2626 y SDict begin H.R end 2674 2626 a 2674 2690 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.7) cvn H.B /ANN pdfmark end 2674 2690 a Fw(\))f(and)f(\()2978 2690 y SDict begin H.S end 2978 2690 a Fw(5.8)3103 2626 y SDict begin H.R end 3103 2626 a 3103 2690 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.8) cvn H.B /ANN pdfmark end 3103 2690 a Fw(\).)53 b(Denote)35 b(b)m(y)94 2806 y Fu(I)8 b(n)p Fw(\()p Fu(k)s Fw(\))43 b(\(resp.)73 b Fu(O)s(ut)p Fw(\()p Fu(k)s Fw(\)\))41 b(the)h(set)h(of)f(v)m(ertices)h Fu(j)49 b Fw(\(resp.)43 b Fu(i)p Fw(\))f(suc)m(h)i(that)e(dim)15 b Fu(A)3073 2821 y Fq(k)r(;j)3212 2806 y Fw(=)44 b(1)d(\(resp.)94 2922 y(dim)16 b Fu(A)347 2937 y Fq(i;k)461 2922 y Fw(=)27 b(1\).)43 b(In)33 b(view)g(of)f(\()1217 2922 y SDict begin H.S end 1217 2922 a Fw(9.2)1341 2858 y SDict begin H.R end 1341 2858 a 1341 2922 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.9.2) cvn H.B /ANN pdfmark end 1341 2922 a Fw(\),)h(ev)m(ery)h(arro)m(w)e Fu(a)h Fw(with)f(b)s(oth)g(ends)h(in)f Fu(I)8 b(n)p Fw(\()p Fu(k)s Fw(\))22 b Fr([)g Fu(O)s(ut)p Fw(\()p Fu(k)s Fw(\))31 b(has)94 3038 y Fu(h)p Fw(\()p Fu(a)p Fw(\))38 b Fr(2)g Fu(I)8 b(n)p Fw(\()p Fu(k)s Fw(\))38 b(and)g Fu(t)p Fw(\()p Fu(a)p Fw(\))f Fr(2)h Fu(O)s(ut)p Fw(\()p Fu(k)s Fw(\).)59 b(W)-8 b(e)39 b(denote)g(the)f (set)h(of)f(these)h(arro)m(ws)g(b)m(y)g Fu(Q)3312 3002 y Fn(0)3312 3063 y Fp(1)3352 3038 y Fw(.)60 b(Since)94 3169 y(\()132 3169 y SDict begin H.S end 132 3169 a Fw(9.1)257 3105 y SDict begin H.R end 257 3105 a 257 3169 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.9.1) cvn H.B /ANN pdfmark end 257 3169 a Fw(\))33 b(holds)f(for)754 3144 y Fm(e)732 3169 y Fu(A)p Fw(,)h(the)g(arro)m(ws)g(of)1478 3144 y Fm(e)1456 3169 y Fu(A)f Fw(can)h(b)s(e)g(unam)m(biguously)f (denoted)h(as)g(follo)m(ws:)318 3315 y Fr(\017)39 b Fm(e)-53 b Fu(a)460 3330 y Fq(i;j)568 3315 y Fw(=)28 b Fu(a)723 3330 y Fq(i;j)835 3315 y Fw(for)33 b(all)d Fu(i;)17 b(j)38 b Fw(di\013eren)m(t)33 b(from)f Fu(k)s Fw(.)318 3431 y Fr(\017)39 b Fm(e)-53 b Fu(a)460 3446 y Fq(i;j)568 3431 y Fw(=)28 b([)p Fu(a)750 3446 y Fq(i;k)836 3431 y Fu(a)887 3446 y Fq(k)r(;j)982 3431 y Fw(])33 b(for)f(all)e Fu(i)e Fr(2)g Fu(O)s(ut)p Fw(\()p Fu(k)s Fw(\))p Fu(;)17 b(j)33 b Fr(2)28 b Fu(I)8 b(n)p Fw(\()p Fu(k)s Fw(\).)318 3547 y Fr(\017)39 b Fm(e)-53 b Fu(a)460 3562 y Fq(j;k)579 3547 y Fw(=)27 b Fu(a)733 3511 y Fq(?)733 3573 y(k)r(;j)861 3547 y Fw(for)32 b Fu(j)h Fr(2)28 b Fu(I)8 b(n)p Fw(\()p Fu(k)s Fw(\).)318 3669 y Fr(\017)39 b Fm(e)-53 b Fu(a)460 3684 y Fq(k)r(;i)574 3669 y Fw(=)28 b Fu(a)729 3633 y Fq(?)729 3695 y(i;k)848 3669 y Fw(for)k Fu(i)c Fr(2)g Fu(O)s(ut)p Fw(\()p Fu(k)s Fw(\).)194 3816 y(W)-8 b(e)33 b(can)g(split)e Fu(S)39 b Fw(in)m(to)31 b(the)i(sum)g(of)f(four)g (terms)94 3992 y(\(9.4\))1387 3992 y SDict begin H.S end 1387 3992 a 1387 3992 a SDict begin 14 H.A end 1387 3992 a 1387 3992 a SDict begin [ /View [/XYZ H.V] /Dest (equation.9.4) cvn H.B /DEST pdfmark end 1387 3992 a Fu(S)h Fw(=)28 b Fu(S)1644 4007 y Fp(1)1705 3992 y Fw(+)22 b Fu(S)1863 4007 y Fp(2)1925 3992 y Fw(+)g Fu(S)2083 4007 y Fp(3)2145 3992 y Fw(+)g Fu(S)2303 4007 y Fp(4)2342 3992 y Fu(;)94 4168 y Fw(where)318 4314 y Fr(\017)41 b Fu(S)469 4329 y Fp(1)541 4314 y Fw(in)m(v)m(olv)m(es)33 b(\(orien)m(ted\))g(3-cycles) g Fu(a)1772 4329 y Fq(i;k)1858 4314 y Fu(a)1909 4329 y Fq(k)r(;j)2004 4314 y Fu(a)2055 4329 y Fq(j;i)2132 4314 y Fw(.)318 4430 y Fr(\017)41 b Fu(S)469 4445 y Fp(2)541 4430 y Fw(in)m(v)m(olv)m(es)33 b(c)m(hordless)h Fu(d)p Fw(-cycles)f(through)f Fu(k)k Fw(with)c Fu(d)c Fr(\025)g Fw(4;)318 4546 y Fr(\017)41 b Fu(S)469 4561 y Fp(3)549 4546 y Fw(in)m(v)m(olv)m(es)g(c)m(hordless)g(cycles)h(ha)m(ving)e(an)g (arro)m(w)h Fu(a)2432 4561 y Fq(j;i)2548 4546 y Fw(for)f Fu(i)h Fr(2)h Fu(O)s(ut)p Fw(\()p Fu(k)s Fw(\))p Fu(;)17 b(j)45 b Fr(2)c Fu(I)8 b(n)p Fw(\()p Fu(k)s Fw(\))409 4663 y(but)33 b(not)f(passing)h(through)f Fu(k)s Fw(;)318 4779 y Fr(\017)41 b Fu(S)469 4794 y Fp(4)536 4779 y Fw(in)m(v)m(olv)m (es)29 b(c)m(hordless)f(cycles)h(ha)m(ving)e(no)h(arro)m(ws)g(with)f(b) s(oth)h(ends)h(in)e Fu(I)8 b(n)p Fw(\()p Fu(k)s Fw(\))k Fr([)g(f)p Fu(k)s Fr(g)g([)409 4895 y Fu(O)s(ut)p Fw(\()p Fu(k)s Fw(\).)94 5041 y(Using)39 b(\()413 5041 y SDict begin H.S end 413 5041 a Fw(9.2)538 4977 y SDict begin H.R end 538 4977 a 538 5041 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.9.2) cvn H.B /ANN pdfmark end 538 5041 a Fw(\),)i(it)d(is)h(easy)h(to)f(see)i(that)e(ev)m(ery)i(c)m(hordless)f (cycle)g Fu(I)47 b Fw(in)m(v)m(olv)m(ed)40 b(in)f Fu(S)3177 5056 y Fp(2)3255 5041 y Fw(or)g Fu(S)3441 5056 y Fp(3)3520 5041 y Fw(has)94 5158 y(exactly)e(one)f(common)e(p)s(oin)m(t)h(with)g (eac)m(h)i(of)e(the)h(sets)h Fu(I)8 b(n)p Fw(\()p Fu(k)s Fw(\))36 b(and)g Fu(O)s(ut)p Fw(\()p Fu(k)s Fw(\).)52 b(Also)35 b(note)h(that)94 5274 y(ev)m(ery)f(term)c(in)h Fu(S)758 5289 y Fp(1)830 5274 y Fw(or)g Fu(S)1009 5289 y Fp(3)1080 5274 y Fw(con)m(tains)h(exactly)f(one)h(arro)m(w)f(from)f Fu(Q)2557 5238 y Fn(0)2557 5298 y Fp(1)2597 5274 y Fw(,)i(while)e(none) i(of)f(the)g(terms)94 5404 y(in)40 b Fu(S)276 5419 y Fp(2)355 5404 y Fw(or)f Fu(S)541 5419 y Fp(4)620 5404 y Fw(con)m(tain)h(suc)m(h)h(arro)m(ws.)66 b(Remem)m(b)s(ering)39 b(\()2236 5404 y SDict begin H.S end 2236 5404 a Fw(5.8)2360 5340 y SDict begin H.R end 2360 5340 a 2360 5404 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.8) cvn H.B /ANN pdfmark end 2360 5404 a Fw(\),)j(w)m(e)f(write)e(the)h(p)s(oten) m(tial)3482 5379 y Fm(e)3469 5404 y Fu(S)45 b Fw(as)94 5520 y(follo)m(ws:)94 5696 y(\(9.5\))1156 5696 y SDict begin H.S end 1156 5696 a 1156 5696 a SDict begin 14 H.A end 1156 5696 a 1156 5696 a SDict begin [ /View [/XYZ H.V] /Dest (equation.9.5) cvn H.B /DEST pdfmark end 1156 5696 a 1170 5671 a Fm(e)1156 5696 y Fu(S)34 b Fw(=)27 b([)p Fu(S)1440 5711 y Fp(1)1480 5696 y Fw(])22 b(+)g([)p Fu(S)1714 5711 y Fp(2)1754 5696 y Fw(])g(+)g([)p Fu(S)1988 5711 y Fp(3)2027 5696 y Fw(])h(+)f([)p Fu(S)2262 5711 y Fp(4)2301 5696 y Fw(])g(+)g(\001)2529 5711 y Fq(k)2572 5696 y Fu(:)p eop end end %%Page: 38 38 TeXDict begin HPSdict begin 38 37 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.38) cvn H.B /DEST pdfmark end 94 218 a Fs(38)492 b(HARM)25 b(DERKSEN,)f(JERZY)i(WEYMAN,)f(AND)f(ANDREI)g(ZELEVINSKY)94 417 y Fw(W)-8 b(e)33 b(ha)m(v)m(e)1332 536 y([)p Fu(S)1419 551 y Fp(1)1458 536 y Fw(])28 b(=)1667 441 y Fm(X)1616 658 y Fq(a)1653 668 y Fg(j;i)1724 658 y Fn(2)p Fq(Q)1827 635 y Ff(0)1827 679 y Fo(1)1878 536 y Fu(x)1933 551 y Fn(f)p Fq(i;j;k)r Fn(g)2136 536 y Fm(e)-53 b Fu(a)2189 551 y Fq(i;j)2270 536 y Fu(a)2321 551 y Fq(j;i)2397 536 y Fu(;)94 817 y Fw(and)33 b(this)f(is)h(the)g(degree)g(2)g(comp)s(onen) m(t)1638 792 y Fm(e)1624 817 y Fu(S)1690 781 y Fp(\(2\))1817 817 y Fw(of)1941 792 y Fm(e)1928 817 y Fu(S)6 b Fw(.)43 b(In)33 b(view)g(of)g(\()2561 817 y SDict begin H.S end 2561 817 a Fw(5.16)2734 753 y SDict begin H.R end 2734 753 a 2734 817 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.16) cvn H.B /ANN pdfmark end 2734 817 a Fw(\),)g(the)g(arro)m(ws)g(in)p 3426 737 74 4 v 32 w Fu(A)f Fw(are)94 947 y(obtained)h(from)e(those)i (in)1117 922 y Fm(e)1095 947 y Fu(A)g Fw(b)m(y)g(remo)m(ving)f(all)e (arro)m(ws)j Fu(a)2257 962 y Fq(j;i)2362 947 y Fr(2)28 b Fu(Q)2533 911 y Fn(0)2533 972 y Fp(1)2605 947 y Fw(and)k(their)h(opp) s(osites)d Fm(e)-53 b Fu(a)3505 962 y Fq(i;j)3585 947 y Fw(.)194 1064 y(Insp)s(ecting)33 b(the)g(other)g(terms)f(in)g(\()1512 1064 y SDict begin H.S end 1512 1064 a Fw(9.5)1637 999 y SDict begin H.R end 1637 999 a 1637 1064 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.9.5) cvn H.B /ANN pdfmark end 1637 1064 a Fw(\),)g(it)g(is)g(easy)i(to)e(see)h(that)1404 1228 y([)p Fu(S)1491 1243 y Fp(2)1530 1228 y Fw(])28 b(=)p 1689 1148 66 4 v 28 w Fu(S)1754 1243 y Fp(3)1794 1228 y Fu(;)114 b Fw([)p Fu(S)2022 1243 y Fp(4)2062 1228 y Fw(])27 b(=)p 2220 1148 V 28 w Fu(S)2286 1243 y Fp(4)2325 1228 y Fu(;)1288 1411 y Fw(\001)1369 1426 y Fq(k)1440 1411 y Fw(=)p 1543 1331 V 27 w Fu(S)1609 1426 y Fp(1)1670 1411 y Fw(+)1819 1317 y Fm(X)1768 1533 y Fq(a)1805 1543 y Fg(j;i)1876 1533 y Fn(2)p Fq(Q)1979 1510 y Ff(0)1979 1554 y Fo(1)2028 1411 y Fm(e)-53 b Fu(a)2081 1426 y Fq(i;j)2159 1411 y Fm(e)g Fu(a)2212 1426 y Fq(j;k)2301 1411 y Fm(e)g Fu(a)2354 1426 y Fq(k)r(;i)2441 1411 y Fu(;)1463 1703 y Fw([)p Fu(S)1550 1718 y Fp(3)1590 1703 y Fw(])28 b(=)1799 1608 y Fm(X)1748 1824 y Fq(a)1785 1834 y Fg(j;i)1856 1824 y Fn(2)p Fq(Q)1959 1802 y Ff(0)1959 1846 y Fo(1)2010 1703 y Fu(f)2058 1718 y Fq(i;j)2138 1703 y Fu(a)2189 1718 y Fq(j;i)2265 1703 y Fu(;)94 1975 y Fw(where)34 b Fu(f)424 1990 y Fq(i;j)532 1975 y Fw(=)28 b Fu(@)687 1990 y Fq(a)724 2000 y Fg(j;i)799 1975 y Fu(S)859 1990 y Fp(3)899 1975 y Fw(,)k(and)h(the)g(terms)p 1587 1895 V 32 w Fu(S)1653 1990 y Fp(1)1692 1975 y Fu(;)p 1736 1895 V 17 w(S)1802 1990 y Fp(3)1873 1975 y Fw(and)p 2063 1895 V 33 w Fu(S)2129 1990 y Fp(4)2201 1975 y Fw(ha)m(v)m(e)g(the)g (same)g(meaning)e(as)h(in)g(\()3499 1975 y SDict begin H.S end 3499 1975 a Fw(9.4)3623 1910 y SDict begin H.R end 3623 1910 a 3623 1975 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.9.4) cvn H.B /ANN pdfmark end 3623 1975 a Fw(\))94 2115 y(with)40 b Fu(A)f Fw(replaced)h(b)m(y)p 970 2035 74 4 v 40 w Fu(A)p Fw(.)64 b(Let)40 b Fu(')f Fw(b)s(e)g(an)h(automorphism)d(of)i Fu(R)q Fr(hh)2646 2090 y Fm(e)2624 2115 y Fu(A)p Fr(ii)f Fw(acting)h(on)g(arro)m(ws)h(as) 94 2231 y(follo)m(ws:)654 2350 y Fu(')p Fw(\()p Fu(a)807 2365 y Fq(j;i)883 2350 y Fw(\))28 b(=)f Fu(x)1107 2309 y Fn(\000)p Fp(1)1107 2381 y Fn(f)p Fq(i;j;k)r Fn(g)1312 2350 y Fw(\()p Fu(a)1401 2365 y Fq(j;i)1500 2350 y Fr(\000)21 b Fm(e)-53 b Fu(a)1651 2365 y Fq(j;k)1739 2350 y Fm(e)h Fu(a)1793 2365 y Fq(k)r(;i)1879 2350 y Fw(\))p Fu(;)114 b(')p Fw(\()n Fm(e)-53 b Fu(a)2211 2365 y Fq(i;j)2291 2350 y Fw(\))28 b(=)d Fm(e)-52 b Fu(a)2512 2365 y Fq(i;j)2614 2350 y Fr(\000)23 b Fu(x)2769 2309 y Fn(\000)p Fp(1)2769 2381 y Fn(f)p Fq(i;j;k)r Fn(g)2974 2350 y Fu(f)3022 2365 y Fq(i;j)94 2502 y Fw(for)33 b(ev)m(ery)h Fu(a)552 2517 y Fq(j;i)656 2502 y Fr(2)28 b Fu(Q)827 2466 y Fn(0)827 2527 y Fp(1)867 2502 y Fw(,)k(and)h Fu(')g Fw(preserv)m(es)i(the)e (rest)g(of)f(the)h(arro)m(ws.)44 b(Then)34 b(w)m(e)g(ha)m(v)m(e)428 2686 y Fu(')p Fw(\()543 2661 y Fm(e)530 2686 y Fu(S)6 b Fw(\))27 b(=)p 765 2606 66 4 v 28 w Fu(S)831 2701 y Fp(1)892 2686 y Fw(+)p 990 2606 V 22 w Fu(S)1056 2701 y Fp(3)1118 2686 y Fw(+)p 1216 2606 V 22 w Fu(S)1282 2701 y Fp(4)1343 2686 y Fw(+)22 b Fu(')p Fw(\()1594 2591 y Fm(X)1543 2808 y Fq(a)1580 2818 y Fg(j;i)1651 2808 y Fn(2)p Fq(Q)1754 2785 y Ff(0)1754 2829 y Fo(1)1788 2686 y Fw(\()p Fu(x)1881 2701 y Fn(f)p Fq(i;j;k)r Fn(g)2084 2686 y Fm(e)-53 b Fu(a)2137 2701 y Fq(i;j)2218 2686 y Fu(a)2269 2701 y Fq(j;i)2367 2686 y Fw(+)20 b Fm(e)-53 b Fu(a)2516 2701 y Fq(i;j)2595 2686 y Fm(e)g Fu(a)2648 2701 y Fq(j;k)2736 2686 y Fm(e)h Fu(a)2790 2701 y Fq(k)r(;i)2898 2686 y Fw(+)22 b Fu(f)3044 2701 y Fq(i;j)3124 2686 y Fu(a)3175 2701 y Fq(j;i)3252 2686 y Fw(\)\))661 2981 y(=)p 765 2901 V 28 w Fu(S)831 2996 y Fp(1)892 2981 y Fw(+)p 990 2901 V 22 w Fu(S)1056 2996 y Fp(3)1118 2981 y Fw(+)p 1216 2901 V 22 w Fu(S)1282 2996 y Fp(4)1343 2981 y Fw(+)1492 2886 y Fm(X)1441 3103 y Fq(a)1478 3113 y Fg(j;i)1549 3103 y Fn(2)p Fq(Q)1652 3080 y Ff(0)1652 3124 y Fo(1)1686 2981 y Fw(\()n Fm(e)-53 b Fu(a)1775 2996 y Fq(i;j)1855 2981 y Fu(a)1906 2996 y Fq(j;i)2005 2981 y Fr(\000)23 b Fu(x)2160 2940 y Fn(\000)p Fp(1)2160 3012 y Fn(f)p Fq(i;j;k)r Fn(g)2365 2981 y Fu(f)2413 2996 y Fq(i;j)2491 2981 y Fm(e)-53 b Fu(a)2544 2996 y Fq(j;k)2633 2981 y Fm(e)g Fu(a)2686 2996 y Fq(k)r(;i)2773 2981 y Fw(\))p Fu(:)94 3283 y Fw(W)-8 b(e)33 b(see)h(that)f(the)g(degree)g(2)f (comp)s(onen)m(t)h(of)f Fu(')p Fw(\()1909 3257 y Fm(e)1896 3283 y Fu(S)6 b Fw(\))32 b(is)1389 3464 y Fu(')p Fw(\()1504 3438 y Fm(e)1491 3464 y Fu(S)5 b Fw(\))1594 3422 y Fp(\(2\))1716 3464 y Fw(=)1870 3369 y Fm(X)1820 3585 y Fq(a)1857 3595 y Fg(j;i)1928 3585 y Fn(2)p Fq(Q)2031 3563 y Ff(0)2031 3607 y Fo(1)2079 3464 y Fm(e)-53 b Fu(a)2132 3479 y Fq(i;j)2213 3464 y Fu(a)2264 3479 y Fq(j;i)2340 3464 y Fu(;)94 3737 y Fw(and)33 b(a)f(momen)m(t's)h(re\015ection)f(sho)m(ws)j(that)1366 3897 y Fr(\000)1511 3802 y Fm(X)1460 4019 y Fq(a)1497 4029 y Fg(j;i)1568 4019 y Fn(2)p Fq(Q)1671 3996 y Ff(0)1671 4040 y Fo(1)1722 3897 y Fu(x)1777 3855 y Fn(\000)p Fp(1)1777 3928 y Fn(f)p Fq(i;j;k)r Fn(g)1982 3897 y Fu(f)2030 3912 y Fq(i;j)2108 3897 y Fm(e)-53 b Fu(a)2161 3912 y Fq(j;k)2250 3897 y Fm(e)g Fu(a)2303 3912 y Fq(k)r(;i)94 4189 y Fw(can)31 b(b)s(e)f(view)m(ed)h(as)f(the)h(comp)s(onen)m(t)p 1497 4109 V 30 w Fu(S)1563 4204 y Fp(2)1632 4189 y Fw(of)e(a)h(primitiv)m(e) e(p)s(oten)m(tial)g(on)p 2776 4109 74 4 v 30 w Fu(A)p Fw(.)43 b(W)-8 b(e)30 b(conclude)h(that)94 4305 y Fu(\026)153 4320 y Fq(k)196 4305 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\))32 b(is)g(righ)m(t-equiv)-5 b(alen)m(t)31 b(to)h(\()p 1441 4225 V Fu(A;)p 1558 4225 66 4 v 17 w(S)1624 4320 y Fp(1)1685 4305 y Fw(+)p 1783 4225 V 22 w Fu(S)1849 4320 y Fp(2)1911 4305 y Fw(+)p 2009 4225 V 22 w Fu(S)2075 4320 y Fp(3)2136 4305 y Fw(+)p 2234 4225 V 22 w Fu(S)2300 4320 y Fp(4)2339 4305 y Fw(\),)h(\014nishing)f(the)h(pro)s(of.)338 b Fh(\003)194 4474 y Fw(W)-8 b(e)36 b(conclude)f(this)g(section)g(b)m(y)h(brie\015y)f (discussing)h(a)e(connection)i(b)s(et)m(w)m(een)h(Jacobian)d(al-)94 4591 y(gebras)45 b(of)f(rigid)e(QPs)j(and)g(cluster-tilted)e(algebras)g (in)m(tro)s(duced)i(in)e([)2829 4591 y SDict begin H.S end 2829 4591 a Fw(8)2878 4526 y SDict begin H.R end 2878 4526 a 2878 4591 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.BMR1) cvn H.B /ANN pdfmark end 2878 4591 a Fw(].)79 b(W)-8 b(e)45 b(refer)f(to)g([)3585 4591 y SDict begin H.S end 3585 4591 a Fw(8)3634 4526 y SDict begin H.R end 3634 4526 a 3634 4591 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.BMR1) cvn H.B /ANN pdfmark end 3634 4591 a Fw(])94 4707 y(for)d(precise)h(de\014nitions;)i(roughly)d(sp)s(eaking,)i (cluster-tilted)c(algebras)i(are)g(endomorphism)94 4823 y(rings)e(of)f(tilting)e(ob)5 b(jects)40 b(in)e(cluster)i(categories.) 62 b(One)39 b(can)g(asso)s(ciate)g(suc)m(h)h(an)f(algebra)f(to)94 4939 y(an)m(y)29 b(quiv)m(er)g(whic)m(h)g(is)e(m)m(utation-equiv)-5 b(alen)m(t)26 b(to)i(a)f(Dynkin)h(quiv)m(er,)i(and)e(these)i(algebras)d (w)m(ere)94 5056 y(explicitly)38 b(computed)j(in)e(terms)g(of)h (generators)g(and)g(relations)e(in)h([)2727 5056 y SDict begin H.S end 2727 5056 a Fw(13)2825 4991 y SDict begin H.R end 2825 4991 a 2825 5056 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.CCS) cvn H.B /ANN pdfmark end 2825 5056 a Fw(,)j(Theorem)e(4.1])f(and)94 5172 y([)121 5172 y SDict begin H.S end 121 5172 a Fw(9)170 5108 y SDict begin H.R end 170 5108 a 170 5172 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.BMR2) cvn H.B /ANN pdfmark end 170 5172 a Fw(].)53 b(Comparing)35 b(their)g(description)g(with)h(that)f (of)g(the)i(Jacobian)e(algebra)f(asso)s(ciated)i(with)94 5288 y(a)d(primitiv)m(e)d(p)s(oten)m(tial,)h(w)m(e)j(see)f(that)g(Prop) s(osition)2070 5288 y SDict begin H.S end 2070 5288 a Fw(9.1)2194 5224 y SDict begin H.R end 2194 5224 a 2194 5288 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.9.1) cvn H.B /ANN pdfmark end 2194 5288 a 33 w Fw(has)g(the)g(follo)m(wing)d (corollary)-8 b(.)94 5307 y SDict begin H.S end 94 5307 a 94 5307 a SDict begin 14 H.A end 94 5307 a 94 5307 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.9.3) cvn H.B /DEST pdfmark end 94 5307 a 157 x FA(Corollary)50 b(9.3.)e Fv(If)c(a)g(quiver)h Fu(Q)g Fv(with)g(the)g(arr)-5 b(ow)44 b(sp)-5 b(an)44 b Fu(A)h Fv(is)f(mutation-e)-5 b(quivalent)45 b(to)g(a)94 5580 y(Dynkin)29 b(quiver,)h(then)f(the)g(Jac)-5 b(obian)28 b(algebr)-5 b(a)28 b Fr(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\))29 b Fv(of)g(the)g(rigid)g(QP)g(on)g Fu(A)g Fv(is)g(isomorphic)94 5696 y(to)36 b(the)e(cluster-tilte)-5 b(d)36 b(algebr)-5 b(a)34 b(asso)-5 b(ciate)g(d)33 b(to)i Fu(Q)p Fv(.)p eop end end %%Page: 39 39 TeXDict begin HPSdict begin 39 38 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.39) cvn H.B /DEST pdfmark end 94 218 a 1179 w Fs(QUIVERS)31 b(WITH)i(POTENTIALS)g(I)1102 b(39)194 417 y Fw(Th)m(us,)42 b(the)d(Jacobian)f(algebras)g(of)g(QPs)h(can)g(b)s (e)g(view)m(ed)h(as)f(generalizations)d(of)j(cluster-)94 533 y(tilted)32 b(algebras.)94 691 y SDict begin H.S end 94 691 a 94 691 a SDict begin 14 H.A end 94 691 a 94 691 a SDict begin [ /View [/XYZ H.V] /Dest (section.10) cvn H.B /DEST pdfmark end 94 691 a 557 788 a Fw(10.)49 b Fx(Decora)-7 b(ted)37 b(represent)-7 b(a)g(tions)36 b(and)i(their)f(mut)-7 b(a)g(tions)194 962 y Fw(The)34 b(follo)m(wing)c(de\014nition)h(is)h (inspired)g(b)m(y)i([)1870 962 y SDict begin H.S end 1870 962 a Fw(28)1967 898 y SDict begin H.R end 1967 898 a 1967 962 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.mrz) cvn H.B /ANN pdfmark end 1967 962 a Fw(].)94 987 y SDict begin H.S end 94 987 a 94 987 a SDict begin 14 H.A end 94 987 a 94 987 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.10.1) cvn H.B /DEST pdfmark end 94 987 a 161 x FA(De\014nition)i (10.1.)41 b Fw(A)32 b Fv(de)-5 b(c)g(or)g(ate)g(d)33 b(r)-5 b(epr)g(esentation)39 b Fw(of)31 b(a)g(QP)h(\()p Fu(A;)17 b(S)6 b Fw(\))32 b(is)f(a)g(pair)g Fr(M)c Fw(=)h(\()p Fu(M)5 b(;)17 b(V)k Fw(\),)94 1264 y(where)54 b Fu(V)74 b Fw(is)52 b(a)g(\014nite-dimensional)d(\(left\))i Fu(R)q Fw(-mo)s(dule,)56 b(and)d Fu(M)63 b Fw(is)51 b(a)i (\014nite-dimensional)94 1380 y Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fw(-mo)s(dule)30 b(annihilated)h(b)m(y)i Fu(J)9 b Fw(\()p Fu(S)d Fw(\))33 b(and)f(b)m(y)i Fl(m)2058 1344 y Fq(n)2138 1380 y Fw(for)e Fu(n)c Fr(\035)f Fw(0.)194 1566 y(Equiv)-5 b(alen)m(tly)d(,)31 b(w)m(e)i(view)e Fu(M)43 b Fw(as)31 b(a)g Fr(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\)-mo)s(dule)30 b(annihilated)f(b)m(y)k(the)e(image)f(of)h Fl(m)3466 1530 y Fq(n)3545 1566 y Fw(for)94 1682 y Fu(n)d Fr(\035)g Fw(0.)42 b(W)-8 b(e)31 b(will)e(sometimes)g(write)i Fr(M)c Fw(=)h(\()p Fu(A;)17 b(S;)g(M)5 b(;)17 b(V)k Fw(\))31 b(and)f(refer)h(to)g(suc)m(h)h(a)e(quadruple)h(as)94 1799 y(a)i Fv(QP-r)-5 b(epr)g(esentation)p Fw(.)194 1915 y(W)d(e)28 b(ha)m(v)m(e)h Fu(M)39 b Fw(=)813 1840 y Fm(L)924 1944 y Fq(i)p Fn(2)p Fq(Q)1051 1953 y Fo(0)1106 1915 y Fu(M)1200 1930 y Fq(i)1256 1915 y Fw(and)28 b Fu(V)49 b Fw(=)1650 1840 y Fm(L)1761 1944 y Fq(i)p Fn(2)p Fq(Q)1888 1953 y Fo(0)1943 1915 y Fu(V)2000 1930 y Fq(i)2028 1915 y Fw(,)29 b(where)g Fu(M)2455 1930 y Fq(i)2511 1915 y Fw(=)e Fu(e)2659 1930 y Fq(i)2688 1915 y Fu(M)38 b Fw(and)28 b Fu(V)3062 1930 y Fq(i)3118 1915 y Fw(=)f Fu(e)3266 1930 y Fq(i)3295 1915 y Fu(V)21 b Fw(.)42 b(With)94 2040 y(some)28 b(abuse)h(of)e(notation,)h(for)f Fu(u)g Fr(2)h Fu(R)q Fr(hh)p Fu(A)p Fr(ii)f Fw(or)g Fu(u)h Fr(2)g(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\),)28 b(w)m(e)h(denote)g(the)f(m)m (ultiplication)94 2156 y(op)s(erator)45 b Fu(m)j Fr(7!)g Fu(um)d Fw(on)g Fu(M)55 b Fw(simply)44 b(as)h Fu(u)j Fw(:)g Fu(M)59 b Fr(!)48 b Fu(M)10 b Fw(;)52 b(w)m(e)46 b(write)e Fu(u)k Fw(=)h Fu(u)3086 2171 y Fq(M)3209 2156 y Fw(if)43 b(w)m(e)j(need)94 2272 y(to)38 b(emphasize)f(the)h(dep)s (endence)i(of)e(this)f(op)s(erator)g(on)g Fu(M)10 b Fw(.)60 b(In)37 b(particular,)h(for)f(eac)m(h)h(arro)m(w)94 2389 y Fu(a)28 b Fr(2)g Fu(A)p Fw(,)33 b(w)m(e)h(ha)m(v)m(e)g Fu(a)28 b Fw(:)f Fu(M)996 2404 y Fq(t)p Fp(\()p Fq(a)p Fp(\))1146 2389 y Fr(!)h Fu(M)1368 2404 y Fq(h)p Fp(\()p Fq(a)p Fp(\))1505 2389 y Fw(,)33 b(and)g Fu(a)p Fr(j)1834 2404 y Fq(M)1902 2414 y Fg(i)1959 2389 y Fw(=)28 b(0)k(for)g Fu(i)c Fr(6)p Fw(=)g Fu(t)p Fw(\()p Fu(a)p Fw(\).)194 2505 y(The)f(aim)c(of)i(this)g(section)h(is)f(to)g(extend)i(the)f (de\014nition)e(of)h(QP-m)m(utations)f(in)h(Corollary)3537 2505 y SDict begin H.S end 3537 2505 a Fw(5.4)3661 2441 y SDict begin H.R end 3661 2441 a 3661 2505 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.5.4) cvn H.B /ANN pdfmark end 3661 2505 a 94 2621 a Fw(and)31 b(De\014nition)734 2621 y SDict begin H.S end 734 2621 a Fw(5.5)859 2557 y SDict begin H.R end 859 2557 a 859 2621 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.5.5) cvn H.B /ANN pdfmark end 859 2621 a 31 w Fw(to)f(the)h(lev)m(el)f(of)g(QP-represen)m(tations,) j(and)d(to)h(pro)m(v)m(e)h(a)e(represen)m(tation-)94 2737 y(theoretic)k(extension)h(of)f(Theorem)1461 2737 y SDict begin H.S end 1461 2737 a Fw(5.7)1585 2673 y SDict begin H.R end 1585 2673 a 1585 2737 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.5.7) cvn H.B /ANN pdfmark end 1585 2737 a Fw(.)48 b(T)-8 b(o)34 b(do)g(this,)g(w)m(e)h(\014rst)f (in)m(tro)s(duce)g(righ)m(t-equiv)-5 b(alence)94 2853 y(for)33 b(QP-represen)m(tations.)94 2873 y SDict begin H.S end 94 2873 a 94 2873 a SDict begin 14 H.A end 94 2873 a 94 2873 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.10.2) cvn H.B /DEST pdfmark end 94 2873 a 166 x FA(De\014nition)f(10.2.) 39 b Fw(Let)29 b(\()p Fu(A;)17 b(S)6 b Fw(\))28 b(and)h(\()p Fu(A)1637 3003 y Fn(0)1660 3039 y Fu(;)17 b(S)1770 3003 y Fn(0)1793 3039 y Fw(\))29 b(b)s(e)g(QPs)g(on)g(the)g(same)g(set)g(of) f(v)m(ertices,)j(and)e(let)94 3155 y Fr(M)37 b Fw(=)f(\()p Fu(M)5 b(;)17 b(V)22 b Fw(\))38 b(\(resp.)g Fr(M)1095 3119 y Fn(0)1155 3155 y Fw(=)f(\()p Fu(M)1410 3119 y Fn(0)1434 3155 y Fu(;)17 b(V)1556 3119 y Fn(0)1579 3155 y Fw(\)\))38 b(b)s(e)g(a)f(decorated)i(represen)m(tation)g(of)e(\()p Fu(A;)17 b(S)6 b Fw(\))37 b(\(resp.)94 3272 y(of)c(\()p Fu(A)317 3236 y Fn(0)340 3272 y Fu(;)17 b(S)450 3236 y Fn(0)473 3272 y Fw(\)\).)43 b(A)33 b Fv(right-e)-5 b(quivalenc)g(e)38 b Fw(b)s(et)m(w)m(een)d Fr(M)d Fw(and)h Fr(M)2307 3236 y Fn(0)2362 3272 y Fw(is)f(a)h(triple)e(\()p Fu(';)17 b( )t(;)g(\021)t Fw(\),)31 b(where:)318 3418 y Fr(\017)41 b Fu(')28 b Fw(:)g Fu(R)q Fr(hh)p Fu(A)p Fr(ii)f(!)g Fu(R)q Fr(hh)p Fu(A)1240 3382 y Fn(0)1263 3418 y Fr(ii)32 b Fw(is)g(a)g(righ)m(t-equiv)-5 b(alence)32 b(b)s(et)m(w)m(een)j(\()p Fu(A;)17 b(S)6 b Fw(\))32 b(and)h(\()p Fu(A)3272 3382 y Fn(0)3295 3418 y Fu(;)17 b(S)3405 3382 y Fn(0)3428 3418 y Fw(\))32 b(\(see)409 3534 y(De\014nition)863 3534 y SDict begin H.S end 863 3534 a Fw(4.2)988 3470 y SDict begin H.R end 988 3470 a 988 3534 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.2) cvn H.B /ANN pdfmark end 988 3534 a Fw(\);)318 3651 y Fr(\017)41 b Fu( )c Fw(:)c Fu(M)43 b Fr(!)33 b Fu(M)943 3615 y Fn(0)1003 3651 y Fw(is)i(a)g(v)m(ector)i(space)g(isomorphism)c(suc)m(h)k(that)e Fu( )28 b Fr(\016)c Fu(u)2979 3666 y Fq(M)3091 3651 y Fw(=)33 b Fu(')p Fw(\()p Fu(u)p Fw(\))3396 3666 y Fq(M)3471 3647 y Ff(0)3520 3651 y Fr(\016)24 b Fu( )409 3767 y Fw(for)32 b Fu(u)27 b Fr(2)h Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fw(;)318 3883 y Fr(\017)41 b Fu(\021)31 b Fw(:)d Fu(V)49 b Fr(!)28 b Fu(V)855 3847 y Fn(0)911 3883 y Fw(is)k(an)h(isomorphism)d (of)i Fu(R)q Fw(-mo)s(dules.)94 3972 y SDict begin H.S end 94 3972 a 94 3972 a SDict begin 14 H.A end 94 3972 a 94 3972 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.10.3) cvn H.B /DEST pdfmark end 94 3972 a 97 x FA(Remark)j(10.3.)k Fw(The)32 b(usual)d(notion)g(of)h(isomorphism)d(of)j(decorated)h (represen)m(tations)g Fr(M)d Fw(=)94 4185 y(\()p Fu(M)5 b(;)17 b(V)22 b Fw(\))32 b(and)h Fr(M)734 4149 y Fn(0)784 4185 y Fw(=)28 b(\()p Fu(M)1030 4149 y Fn(0)1054 4185 y Fu(;)17 b(V)1176 4149 y Fn(0)1200 4185 y Fw(\))32 b(of)g(the)h(same)f (QP)g(\()p Fu(A;)17 b(S)6 b Fw(\))32 b(\(namely)-8 b(,)32 b(that)g Fu(M)43 b Fw(and)33 b Fu(M)3296 4149 y Fn(0)3352 4185 y Fw(are)g(iso-)94 4301 y(morphic)26 b Fr(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\)-mo)s(dules,)26 b(and)h Fu(V)48 b Fw(and)27 b Fu(V)1792 4265 y Fn(0)1842 4301 y Fw(are)g(isomorphic)d Fu(R)q Fw(-mo)s(dules\))h(is)i(a)f(sp)s(ecial)f (case)94 4418 y(of)30 b(righ)m(t-equiv)-5 b(alence)29 b(with)h(the)h Fu(')p Fw(-comp)s(onen)m(t)f(b)s(eing)f(the)i(iden)m (tit)m(y)-8 b(.)42 b(The)31 b(righ)m(t-equiv)-5 b(alence)94 4534 y(seems)30 b(to)d(b)s(e)i(more)e(relev)-5 b(an)m(t)28 b(for)g(applications)e(to)i(cluster)g(algebras.)42 b(T)-8 b(o)28 b(illustrate,)f(consider)94 4650 y(the)43 b Fv(Kr)-5 b(one)g(cker)43 b(quiver)52 b Fu(Q)42 b Fw(with)g(t)m(w)m(o)g(v)m (ertices)i(1)d(and)h(2,)j(and)d(t)m(w)m(o)g(arro)m(ws)h Fu(a)f Fw(and)g Fu(b)h Fw(from)94 4766 y(1)f(to)f(2.)71 b(Since)42 b Fu(Q)g Fw(has)g(no)f(orien)m(ted)h(cycles,)j(it)c(has)h (only)f(one)h(QP)g(\()p Fu(A;)17 b(S)6 b Fw(\):)62 b(the)42 b(one)g(with)94 4883 y Fu(S)h Fw(=)37 b(0.)59 b(Th)m(us,)40 b(a)e(decorated)h(represen)m(tation)f Fr(M)f Fw(=)f(\()p Fu(M)5 b(;)17 b(V)22 b Fw(\))38 b(with)f Fu(V)58 b Fw(=)37 b(0)g(is)h(just)g(a)g(usual)94 4999 y(represen)m(tation)33 b(of)e(the)g(quiv)m(er)i Fu(Q)p Fw(,)f(i.e.,)f(it)f(consists)j(of)d(t)m (w)m(o)j(v)m(ector)f(spaces)h Fu(M)3052 5014 y Fp(1)3123 4999 y Fw(and)f Fu(M)3406 5014 y Fp(2)3446 4999 y Fw(,)f(and)94 5115 y(t)m(w)m(o)43 b(linear)c(maps)j Fu(a)f Fw(and)h Fu(b)g Fw(from)e Fu(M)1539 5130 y Fp(1)1620 5115 y Fw(to)h Fu(M)1842 5130 y Fp(2)1882 5115 y Fw(.)70 b(Suc)m(h)43 b(represen)m(tations)f(w)m(ere)h(classi\014ed)e(b)m(y)94 5231 y(Kronec)m(k)m(er.)j(In)28 b(particular,)f(he)h(pro)m(v)m(ed)i (that,)e(for)f(ev)m(ery)j Fu(n)e Fr(\025)g Fw(1,)h(the)f(isomorphism)d (classes)k(of)94 5347 y(indecomp)s(osable)h Fu(Q)p Fw(-represen)m (tations)j(with)d(dim)16 b Fu(M)2074 5362 y Fp(1)2141 5347 y Fw(=)28 b(dim)15 b Fu(M)2518 5362 y Fp(2)2585 5347 y Fw(=)28 b Fu(n)j Fw(are)g(naturally)f(param-)94 5464 y(eterized)d(b)m(y)h(the)f(pro)5 b(jectiv)m(e)27 b(line.)40 b(Ho)m(w)m(ev)m(er)29 b(all)24 b(these)k(represen)m(tations) f(are)g(righ)m(t-equiv)-5 b(alen)m(t)94 5580 y(to)32 b(eac)m(h)h(other.)44 b(This)32 b(is)g(more)g(in)f(sync)j(with)d(the)i (fact)f(that,)g(as)h(discussed)h(in)d([)3125 5580 y SDict begin H.S end 3125 5580 a Fw(16)3223 5516 y SDict begin H.R end 3223 5516 a 3223 5580 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.calzel) cvn H.B /ANN pdfmark end 3223 5580 a Fw(],)h(all)e(these)94 5696 y(represen)m(tations)k(giv)m(e)f (rise)f(to)g(the)h(same)g(elemen)m(t)f(of)g(the)h(corresp)s(onding)g (cluster)g(algebra.)p eop end end %%Page: 40 40 TeXDict begin HPSdict begin 40 39 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.40) cvn H.B /DEST pdfmark end 94 218 a Fs(40)492 b(HARM)25 b(DERKSEN,)f(JERZY)i(WEYMAN,)f(AND)f(ANDREI)g(ZELEVINSKY)194 417 y Fw(W)-8 b(e)39 b(no)m(w)h(presen)m(t)g(a)e(represen)m (tation-theoretic)h(extension)h(of)e(Theorem)3033 417 y SDict begin H.S end 3033 417 a Fw(4.6)3158 353 y SDict begin H.R end 3158 353 a 3158 417 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.6) cvn H.B /ANN pdfmark end 3158 417 a Fw(.)62 b(Let)39 b Fr(M)f Fw(=)94 533 y(\()p Fu(A;)17 b(S;)g(M)5 b(;)17 b(V)22 b Fw(\))28 b(b)s(e)g(a)g(QP-represen) m(tation,)h(and)f(let)g Fu(')f Fw(:)h Fu(R)q Fr(hh)p Fu(A)2375 548 y Fp(red)2490 533 y Fr(\010)13 b Fu(C)7 b Fr(ii)27 b(!)h Fu(R)q Fr(hh)p Fu(A)p Fr(ii)f Fw(b)s(e)h(a)g(righ)m (t-)94 649 y(equiv)-5 b(alence)26 b(of)g(the)g(QPs)g(\()p Fu(A)1185 664 y Fp(red)1287 649 y Fu(;)17 b(S)1391 664 y Fp(red)1493 649 y Fw(\))8 b Fr(\010)g Fw(\()p Fu(C)r(;)17 b(T)d Fw(\))25 b(and)h(\()p Fu(A;)17 b(S)6 b Fw(\),)27 b(where)f(\()p Fu(A)2793 664 y Fp(red)2896 649 y Fu(;)17 b(S)3000 664 y Fp(red)3102 649 y Fw(\))25 b(is)g(a)h(reduced)94 766 y(QP)-8 b(,)34 b(and)g(\()p Fu(C)r(;)17 b(T)d Fw(\))33 b(is)g(a)h(trivial)d(QP)-8 b(,)34 b(see)h(Theorem)2016 766 y SDict begin H.S end 2016 766 a Fw(4.6)2141 701 y SDict begin H.R end 2141 701 a 2141 766 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.6) cvn H.B /ANN pdfmark end 2141 766 a Fw(.)47 b(W)-8 b(e)34 b(de\014ne)h(a)f Fu(R)q Fr(hh)p Fu(A)2976 781 y Fp(red)3078 766 y Fr(ii)p Fw(-mo)s(dule)d Fu(M)3637 730 y Fn(0)94 882 y Fw(b)m(y)g(setting)e Fu(M)648 846 y Fn(0)700 882 y Fw(=)e Fu(M)40 b Fw(as)30 b(a)f Fu(K)7 b Fw(-v)m(ector)30 b(space)g(with)f(the)h(action)f(of)g Fu(R)q Fr(hh)p Fu(A)2809 897 y Fp(red)2910 882 y Fr(ii)g Fw(giv)m(en)h(b)m(y)g Fu(u)3457 897 y Fq(M)3532 878 y Ff(0)3586 882 y Fw(=)94 998 y Fu(')p Fw(\()p Fu(u)p Fw(\))290 1013 y Fq(M)369 998 y Fw(.)41 b(In)25 b(view)g(of)f(Prop)s(osition)1388 998 y SDict begin H.S end 1388 998 a Fw(4.5)1513 934 y SDict begin H.R end 1513 934 a 1513 998 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.5) cvn H.B /ANN pdfmark end 1513 998 a Fw(,)i(this)e(mak)m(es)i(a)e(quadruple)h Fr(M)2677 1013 y Fp(red)2807 998 y Fw(=)i(\()p Fu(A)3021 1013 y Fp(red)3124 998 y Fu(;)17 b(S)3228 1013 y Fp(red)3329 998 y Fu(;)g(M)3477 962 y Fn(0)3501 998 y Fu(;)g(V)k Fw(\))94 1114 y(a)33 b(w)m(ell-de\014ned)g(QP-represen)m(tation.)94 1134 y SDict begin H.S end 94 1134 a 94 1134 a SDict begin 14 H.A end 94 1134 a 94 1134 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.10.4) cvn H.B /DEST pdfmark end 94 1134 a 137 x FA(De\014nition)g(10.4.)39 b Fw(W)-8 b(e)30 b(call)e(the)i (QP-represen)m(tation)g Fr(M)2318 1286 y Fp(red)2449 1271 y Fw(giv)m(en)f(b)m(y)i(the)e(ab)s(o)m(v)m(e)i(construc-)94 1388 y(tion)h(the)h Fv(r)-5 b(e)g(duc)g(e)g(d)34 b(p)-5 b(art)43 b Fw(of)32 b Fr(M)p Fw(.)194 1544 y(This)h(terminology)d(is)i (justi\014ed)h(b)m(y)g(the)g(follo)m(wing.)94 1564 y SDict begin H.S end 94 1564 a 94 1564 a SDict begin 14 H.A end 94 1564 a 94 1564 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.10.5) cvn H.B /DEST pdfmark end 94 1564 a 137 x FA(Prop)s(osition)38 b(10.5.)43 b Fv(The)36 b(right-e)-5 b(quivalenc)g(e)35 b(class)h(of)g Fr(M)2388 1716 y Fp(red)2526 1701 y Fv(is)h(determine)-5 b(d)35 b(by)i(the)g(right-)94 1818 y(e)-5 b(quivalenc)g(e)34 b(class)g(of)h Fr(M)p Fv(.)94 1974 y(Pr)-5 b(o)g(of.)42 b Fw(T)-8 b(o)45 b(get)h(more)f(in)g (sync)i(with)f(the)g(notation)e(in)h(Prop)s(osition)2827 1974 y SDict begin H.S end 2827 1974 a Fw(4.9)2952 1910 y SDict begin H.R end 2952 1910 a 2952 1974 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.9) cvn H.B /ANN pdfmark end 2952 1974 a Fw(,)k(w)m(e)e(c)m(hange)g(our)94 2091 y(notation)h(a)g(little)e(bit)i(and)h(assume)g(that)f Fr(M)g Fw(is)h(a)f(decorated)h(represen)m(tation)h(of)e(a)g(QP)94 2207 y(\()p Fu(A)32 b Fr(\010)g Fu(C)r(;)17 b(S)36 b Fw(+)c Fu(T)14 b Fw(\),)49 b(where)e(\()p Fu(A;)17 b(S)6 b Fw(\))46 b(is)g(a)g(reduced)h(QP)-8 b(,)47 b(and)f(\()p Fu(C)r(;)17 b(T)d Fw(\))45 b(a)h(trivial)d(one.)85 b(Let)47 b Fu(')94 2323 y Fw(b)s(e)40 b(an)g(auto-righ)m(t-equiv)-5 b(alence)38 b(of)h(\()p Fu(A)27 b Fr(\010)g Fu(C)r(;)17 b(S)32 b Fw(+)27 b Fu(T)14 b Fw(\),)41 b(that)e(is,)i(an)f(algebra)e (automorphism)94 2439 y(of)48 b Fu(R)q Fr(hh)p Fu(A)32 b Fr(\010)h Fu(C)7 b Fr(ii)46 b Fw(suc)m(h)k(that)d Fu(')p Fw(\()p Fu(S)38 b Fw(+)32 b Fu(T)14 b Fw(\))47 b(is)g(cyclically)f (equiv)-5 b(alen)m(t)48 b(to)f Fu(S)38 b Fw(+)32 b Fu(T)14 b Fw(.)88 b(T)-8 b(o)48 b(pro)m(v)m(e)94 2556 y(Prop)s(osition)619 2556 y SDict begin H.S end 619 2556 a Fw(10.5)793 2491 y SDict begin H.R end 793 2491 a 793 2556 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.10.5) cvn H.B /ANN pdfmark end 793 2556 a Fw(,)32 b(it)g(su\016ces)j(to)d(sho)m(w)i(the)f(follo)m (wing:)259 2713 y SDict begin H.S end 259 2713 a 259 2713 a SDict begin 14 H.A end 259 2713 a 259 2713 a SDict begin [ /View [/XYZ H.V] /Dest (equation.10.1) cvn H.B /DEST pdfmark end 259 2713 a 316 w Fw(there)g(exists)g(an)g(algebra)e (automorphism)g Fu(')2280 2676 y Fn(0)2335 2713 y Fw(of)i Fu(R)q Fr(hh)p Fu(A)p Fr(ii)e Fw(suc)m(h)j(that)-3087 b(\(10.1\))342 2858 y Fu(')406 2822 y Fn(0)429 2858 y Fw(\()p Fu(S)6 b Fw(\))32 b(is)g(cyclically)f(equiv)-5 b(alen)m(t)32 b(to)h Fu(S)6 b Fw(,)32 b(and)h Fu(')2086 2822 y Fn(0)2109 2858 y Fw(\()p Fu(u)p Fw(\))2241 2873 y Fq(M)2347 2858 y Fw(=)28 b Fu(')p Fw(\()p Fu(u)p Fw(\))2647 2873 y Fq(M)2757 2858 y Fw(for)k Fu(u)c Fr(2)g Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fw(.)194 3015 y(As)34 b(in)e(the)h(pro)s(of)f (of)g(Prop)s(osition)1513 3015 y SDict begin H.S end 1513 3015 a Fw(4.9)1637 2951 y SDict begin H.R end 1637 2951 a 1637 3015 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.9) cvn H.B /ANN pdfmark end 1637 3015 a Fw(,)h(let)g Fu(L)g Fw(denote)g(the)h(closure)f(of)f(the)h(t)m(w)m(o-sided)h(ideal) 94 3131 y(in)e Fu(R)q Fr(hh)p Fu(A)22 b Fr(\010)h Fu(C)7 b Fr(ii)32 b Fw(generated)h(b)m(y)h Fu(C)7 b Fw(.)43 b(Recall)31 b(from)g(\()2032 3131 y SDict begin H.S end 2032 3131 a Fw(4.10)2206 3067 y SDict begin H.R end 2206 3067 a 2206 3131 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.4.10) cvn H.B /ANN pdfmark end 2206 3131 a Fw(\))h(that)g(w)m(e)i(ha)m (v)m(e)1302 3288 y Fu(R)q Fr(hh)p Fu(A)22 b Fr(\010)h Fu(C)7 b Fr(ii)27 b Fw(=)g Fu(R)q Fr(hh)p Fu(A)p Fr(ii)22 b(\010)g Fu(L;)94 3445 y Fw(and)1418 3561 y Fu(J)9 b Fw(\()p Fu(S)28 b Fw(+)22 b Fu(T)14 b Fw(\))27 b(=)h Fu(J)9 b Fw(\()p Fu(S)d Fw(\))22 b Fr(\010)h Fu(L)94 3698 y Fw(\(in)37 b(the)g(last)f(equalit)m(y)-8 b(,)37 b Fu(J)9 b Fw(\()p Fu(S)d Fw(\))37 b(is)f(understo)s(o)s(d)i(as)f(the)g (Jacobian)f(ideal)g(of)g Fu(S)42 b Fw(in)37 b Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fw(\).)55 b(In)94 3814 y(particular,)31 b(w)m(e)j(ha)m(v)m(e)g Fu(u)996 3829 y Fq(M)1102 3814 y Fw(=)28 b(0)k(for)g Fu(u)27 b Fr(2)h Fu(L)p Fw(.)194 3930 y(Let)43 b Fu( )48 b Fw(:)c Fu(R)q Fr(hh)p Fu(A)p Fr(ii)f(!)h Fu(R)q Fr(hh)p Fu(A)p Fr(ii)e Fw(denote)h(the)g (restriction)e(to)h Fu(R)q Fr(hh)p Fu(A)p Fr(ii)f Fw(of)h(the)h(comp)s (osition)94 4046 y Fu(p)23 b Fr(\016)g Fu(')p Fw(,)34 b(where)h Fu(p)e Fw(is)g(the)h(pro)5 b(jection)34 b(of)f Fu(R)q Fr(hh)p Fu(A)22 b Fr(\010)h Fu(C)7 b Fr(ii)33 b Fw(on)m(to)h Fu(R)q Fr(hh)p Fu(A)p Fr(ii)f Fw(along)f Fu(L)p Fw(.)47 b(Then)34 b(w)m(e)h(ha)m(v)m(e)94 4162 y Fu( )t Fw(\()p Fu(u)p Fw(\))293 4177 y Fq(M)400 4162 y Fw(=)27 b Fu(')p Fw(\()p Fu(u)p Fw(\))699 4177 y Fq(M)810 4162 y Fw(for)32 b Fu(u)27 b Fr(2)h Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fw(.)194 4279 y(According)g(to)g(\()806 4279 y SDict begin H.S end 806 4279 a Fw(4.12)979 4215 y SDict begin H.R end 979 4215 a 979 4279 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.4.12) cvn H.B /ANN pdfmark end 979 4279 a Fw(\),)h Fu( )j Fw(is)c(an)g(algebra)f(automorphism)f(of)i Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fw(.)41 b(F)-8 b(urthermore,)29 b(using)94 4395 y(\()132 4395 y SDict begin H.S end 132 4395 a Fw(4.12)306 4331 y SDict begin H.R end 306 4331 a 306 4395 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.4.12) cvn H.B /ANN pdfmark end 306 4395 a Fw(\))k(in)f(conjunction)h (with)g(Prop)s(osition)1768 4395 y SDict begin H.S end 1768 4395 a Fw(4.10)1941 4331 y SDict begin H.R end 1941 4331 a 1941 4395 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.10) cvn H.B /ANN pdfmark end 1941 4395 a Fw(,)g(w)m(e)i(conclude)e (that)g Fu(J)9 b Fw(\()p Fu( )t Fw(\()p Fu(S)d Fw(\)\))28 b(=)g Fu(J)9 b Fw(\()p Fu(S)d Fw(\),)33 b(and)94 4511 y(that)25 b(there)g(exists)g(an)g(algebra)e(automorphism)f Fu(\021)29 b Fw(of)24 b Fu(R)q Fr(hh)p Fu(A)p Fr(ii)f Fw(suc)m(h)j(that)f Fu(\021)t Fw(\()p Fu( )t Fw(\()p Fu(S)6 b Fw(\)\))23 b(is)h(cyclically)94 4627 y(equiv)-5 b(alen)m(t)30 b(to)g Fu(S)6 b Fw(,)30 b(and)g Fu(\021)t Fw(\()p Fu(u)p Fw(\))16 b Fr(\000)i Fu(u)27 b Fr(2)h Fu(J)9 b Fw(\()p Fu(S)d Fw(\))30 b(for)f Fu(u)f Fr(2)g Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fw(.)41 b(T)-8 b(aking)30 b Fu(')2772 4591 y Fn(0)2823 4627 y Fw(=)d Fu(\021)21 b Fr(\016)c Fu( )t Fw(,)30 b(w)m(e)h(see)h(that)1054 4784 y Fu(')1118 4743 y Fn(0)1141 4784 y Fw(\()p Fu(u)p Fw(\))1273 4799 y Fq(M)1379 4784 y Fw(=)c Fu(\021)t Fw(\()p Fu( )t Fw(\()p Fu(u)p Fw(\)\))1810 4799 y Fq(M)1915 4784 y Fw(=)g Fu( )t Fw(\()p Fu(u)p Fw(\))2218 4799 y Fq(M)2324 4784 y Fw(=)g Fu(')p Fw(\()p Fu(u)p Fw(\))2624 4799 y Fq(M)94 4941 y Fw(for)33 b Fu(u)27 b Fr(2)h Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fw(.)42 b(Th)m(us)35 b Fu(')1106 4905 y Fn(0)1161 4941 y Fw(satis\014es)f(all)c(the)j(conditions)f(in)f(\() 2442 4941 y SDict begin H.S end 2442 4941 a Fw(10.1)2616 4877 y SDict begin H.R end 2616 4877 a 2616 4941 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.1) cvn H.B /ANN pdfmark end 2616 4941 a Fw(\),)h(and)h(w)m(e)h(are)e(done.)148 b Fh(\003)194 5107 y Fw(W)-8 b(e)28 b(turn)g(to)f(the)h(de\014nition)f (of)g Fv(mutations)36 b Fw(for)27 b(a)g(QP-represen)m(tation)h Fr(M)f Fw(=)h(\()p Fu(A;)17 b(S;)g(M)5 b(;)17 b(V)k Fw(\).)94 5224 y(W)-8 b(e)36 b(\014x)g(a)e(v)m(ertex)j Fu(k)s Fw(,)f(and)f(let)g Fu(a)1291 5239 y Fp(1)1330 5224 y Fu(;)17 b(:)g(:)g(:)f(;)h(a)1600 5239 y Fq(s)1672 5224 y Fw(b)s(e)35 b(all)e(arro)m(ws)i(with)g Fu(h)p Fw(\()p Fu(a)2629 5239 y Fq(p)2669 5224 y Fw(\))d(=)g Fu(k)s Fw(,)j(and)h Fu(b)3197 5239 y Fp(1)3237 5224 y Fu(;)17 b(:)g(:)g(:)e(;)i(b)3496 5239 y Fq(t)3561 5224 y Fw(b)s(e)94 5340 y(all)31 b(arro)m(ws)i(with)f Fu(t)p Fw(\()p Fu(b)878 5355 y Fq(q)917 5340 y Fw(\))c(=)f Fu(k)s Fw(.)44 b(W)-8 b(e)33 b(denote)94 5592 y(\(10.2\))1061 5592 y SDict begin H.S end 1061 5592 a 1061 5592 a SDict begin 14 H.A end 1061 5592 a 1061 5592 a SDict begin [ /View [/XYZ H.V] /Dest (equation.10.2) cvn H.B /DEST pdfmark end 1061 5592 a Fu(M)1155 5607 y Fp(in)1246 5592 y Fw(=)1408 5467 y Fq(s)1349 5497 y Fm(M)1362 5707 y Fq(p)p Fp(=1)1517 5592 y Fu(M)1611 5607 y Fq(t)p Fp(\()p Fq(a)1700 5615 y Fg(p)1738 5607 y Fp(\))1769 5592 y Fu(;)114 b(M)2004 5607 y Fp(out)2139 5592 y Fw(=)2305 5467 y Fq(t)2242 5497 y Fm(M)2255 5707 y Fq(q)r Fp(=1)2409 5592 y Fu(M)2503 5607 y Fq(h)p Fp(\()p Fq(b)2601 5615 y Fg(q)2636 5607 y Fp(\))2668 5592 y Fu(:)p eop end end %%Page: 41 41 TeXDict begin HPSdict begin 41 40 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.41) cvn H.B /DEST pdfmark end 94 218 a 1179 w Fs(QUIVERS)31 b(WITH)i(POTENTIALS)g(I)1102 b(41)94 417 y Fw(Let)36 b Fu(\013)d Fw(=)g Fu(\013)538 432 y Fq(M)649 417 y Fw(:)g Fu(M)803 432 y Fp(in)899 417 y Fr(!)f Fu(M)1125 432 y Fq(k)1203 417 y Fw(and)k Fu(\014)i Fw(=)33 b Fu(\014)1653 432 y Fq(M)1764 417 y Fw(:)g Fu(M)1918 432 y Fq(k)1993 417 y Fr(!)g Fu(M)2220 432 y Fp(out)2361 417 y Fw(b)s(e)j Fu(K)7 b Fw(-linear)34 b(maps)h(giv)m(en)g(in)g(the)94 533 y(matrix)d(form)f(b)m(y)94 881 y(\(10.3\))1085 881 y SDict begin H.S end 1085 881 a 1085 881 a SDict begin 14 H.A end 1085 881 a 1085 881 a SDict begin [ /View [/XYZ H.V] /Dest (equation.10.3) cvn H.B /DEST pdfmark end 1085 881 a Fu(\013)e Fw(=)1279 801 y Fm(\000)1325 879 y Fu(a)1376 894 y Fp(1)1499 879 y Fu(a)1550 894 y Fp(2)1672 879 y Fr(\001)17 b(\001)g(\001)98 b Fu(a)1939 894 y Fq(s)1976 801 y Fm(\001)2038 881 y Fu(;)115 b(\014)33 b Fw(=)2372 621 y Fm(0)2372 797 y(B)2372 856 y(B)2372 920 y(@)2459 683 y Fu(b)2500 698 y Fp(1)2459 799 y Fu(b)2500 814 y Fp(2)2486 894 y Fw(.)2486 927 y(.)2486 960 y(.)2464 1076 y Fu(b)2505 1091 y Fq(t)2540 621 y Fm(1)2540 797 y(C)2540 856 y(C)2540 920 y(A)2643 881 y Fu(:)94 1229 y Fw(W)-8 b(e)40 b(also)e(in)m(tro)s(duce)g(a)h Fu(K)7 b Fw(-linear)37 b(map)h Fu(\015)43 b Fw(=)38 b Fu(\015)1880 1244 y Fq(M)1997 1229 y Fw(:)h Fu(M)2157 1244 y Fp(out)2301 1229 y Fr(!)f Fu(M)2533 1244 y Fp(in)2635 1229 y Fw(as)h(follo)m(ws.)61 b(Replacing)38 b Fu(S)94 1345 y Fw(if)33 b(necessary)i(b)m(y)f(a)f (cyclically)e(equiv)-5 b(alen)m(t)33 b(p)s(oten)m(tial,)f(w)m(e)i(ma)m (y)f(assume)h(that)f Fu(S)h Fr(2)29 b Fu(R)q Fr(hh)p Fu(A)p Fr(ii)3562 1357 y Fp(^)3561 1375 y Fq(k)q(;)3621 1357 y Fp(^)3619 1375 y Fq(k)94 1493 y Fw(\(see)38 b(\()332 1493 y SDict begin H.S end 332 1493 a Fw(6.1)456 1429 y SDict begin H.R end 456 1429 a 456 1493 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.6.1) cvn H.B /ANN pdfmark end 456 1493 a Fw(\)\).)55 b(W)-8 b(e)36 b(iden)m(tify)g Fu(R)q Fr(hh)p Fu(A)p Fr(ii)1448 1505 y Fp(^)1447 1522 y Fq(k)q(;)1507 1505 y Fp(^)1505 1522 y Fq(k)1584 1493 y Fw(with)g Fu(R)q Fr(hh)1985 1468 y Fm(e)1963 1493 y Fu(A)2038 1505 y Fp(^)2036 1522 y Fq(k)q(;)2096 1505 y Fp(^)2094 1522 y Fq(k)2136 1493 y Fr(ii)g Fw(as)g(in)g(Lemma)2842 1493 y SDict begin H.S end 2842 1493 a Fw(6.2)2967 1429 y SDict begin H.R end 2967 1429 a 2967 1493 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.6.2) cvn H.B /ANN pdfmark end 2967 1493 a Fw(.)54 b(This)37 b(allo)m(ws)e(us)94 1613 y(to)e(de\014ne)h(the)f(comp)s(onen)m(t)f Fu(\015)1213 1628 y Fq(p;q)1334 1613 y Fw(:)27 b Fu(M)1482 1628 y Fq(h)p Fp(\()p Fq(b)1580 1636 y Fg(q)1615 1628 y Fp(\))1675 1613 y Fr(!)g Fu(M)1896 1628 y Fq(t)p Fp(\()p Fq(a)1985 1636 y Fg(p)2023 1628 y Fp(\))2087 1613 y Fw(of)32 b Fu(\015)38 b Fw(b)m(y)33 b(setting)94 1770 y(\(10.4\))1580 1770 y SDict begin H.S end 1580 1770 a 1580 1770 a SDict begin 14 H.A end 1580 1770 a 1580 1770 a SDict begin [ /View [/XYZ H.V] /Dest (equation.10.4) cvn H.B /DEST pdfmark end 1580 1770 a Fu(\015)1631 1785 y Fq(p;q)1752 1770 y Fw(=)27 b Fu(@)1906 1786 y Fp([)p Fq(b)1956 1794 y Fg(q)1991 1786 y Fq(a)2028 1794 y Fg(p)2064 1786 y Fp(])2088 1770 y Fu(S:)94 1927 y Fw(Th)m(us,)35 b(w)m(e)e(ha)m(v)m(e)h(constructed)g (the)f(follo)m(wing)d(triangle)h(of)h(linear)f(maps:)94 2091 y(\(10.5\))1331 2091 y SDict begin H.S end 1331 2091 a 1331 2091 a SDict begin 14 H.A end 1331 2091 a 1331 2091 a SDict begin [ /View [/XYZ H.V] /Dest (equation.10.5) cvn H.B /DEST pdfmark end 1331 2091 a 1774 2093 a Fu(M)1868 2108 y Fq(k)2088 2177 y(\014)2194 2319 y Fk(")p Fj(")2162 2293 y Fi(E)2131 2269 y(E)2100 2244 y(E)2069 2219 y(E)2038 2194 y(E)2007 2169 y(E)1976 2145 y(E)1945 2120 y(E)1360 2433 y Fu(M)1454 2448 y Fp(in)1561 2191 y Fq(\013)1745 2120 y Fk(<)p Fj(<)1713 2147 y Fi(z)1684 2171 y(z)1655 2196 y(z)1626 2221 y(z)1596 2245 y(z)1567 2270 y(z)1538 2295 y(z)1508 2319 y(z)2169 2433 y Fu(M)2263 2448 y Fp(out)1833 2485 y Fq(\015)1546 2428 y Fk(o)p Fj(o)p 1546 2429 594 4 v 2398 2091 a Fu(:)94 2568 y SDict begin H.S end 94 2568 a 94 2568 a SDict begin 14 H.A end 94 2568 a 94 2568 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.10.6) cvn H.B /DEST pdfmark end 94 2568 a 94 x FA(Lemma)38 b(10.6.)k Fv(We)35 b(have)f Fu(\013)q(\015)e Fw(=)c(0)34 b Fv(and)h Fu(\015)5 b(\014)33 b Fw(=)27 b(0)p Fv(.)94 2836 y(Pr)-5 b(o)g(of.)42 b Fw(The)33 b Fu(q)t Fw(-th)f(comp)s(onen)m(t)h(of)f Fu(\013)q(\015)37 b Fw(is)32 b(equal)h(to)1431 2914 y Fm(X)1485 3124 y Fq(p)1591 3009 y Fu(a)1642 3024 y Fq(p)1682 3009 y Fu(@)1733 3025 y Fp([)p Fq(b)1783 3033 y Fg(q)1818 3025 y Fq(a)1855 3033 y Fg(p)1891 3025 y Fp(])1915 3009 y Fu(S)h Fw(=)27 b Fu(@)2163 3024 y Fq(b)2193 3032 y Fg(q)2232 3009 y Fu(S)6 b Fw(;)94 3270 y(therefore,)37 b Fu(\013)q(\015)g Fw(=)32 b(0,)k(since)f Fu(M)46 b Fw(is)35 b(a)g(represen)m(tation)h(of)f Fr(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\).)51 b(Similarly)-8 b(,)32 b(the)k Fu(p)p Fw(-th)e(com-)94 3386 y(p)s(onen)m(t)g(of)e Fu(\015)5 b(\014)38 b Fw(is)32 b(equal)g(to)1405 3424 y Fm(X)1460 3633 y Fq(q)1549 3519 y Fw(\()p Fu(@)1638 3534 y Fp([)p Fq(b)1688 3542 y Fg(q)1722 3534 y Fq(a)1759 3542 y Fg(p)1796 3534 y Fp(])1820 3519 y Fu(S)6 b Fw(\))p Fu(b)1965 3534 y Fq(q)2031 3519 y Fw(=)27 b Fu(@)2185 3534 y Fq(a)2222 3542 y Fg(p)2264 3519 y Fu(S;)94 3754 y Fw(implying)j(that)j Fu(\015)5 b(\014)33 b Fw(=)27 b(0.)2551 b Fh(\003)194 3929 y Fw(No)m(w)36 b(let)e(\()623 3903 y Fm(e)600 3929 y Fu(A;)730 3903 y Fm(e)717 3929 y Fu(S)6 b Fw(\))35 b(b)s(e)g(giv)m(en)g(b)m(y)h(\()1424 3929 y SDict begin H.S end 1424 3929 a Fw(5.4)1549 3864 y SDict begin H.R end 1549 3864 a 1549 3929 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.4) cvn H.B /ANN pdfmark end 1549 3929 a Fw(\))f(and)g(\()1852 3929 y SDict begin H.S end 1852 3929 a Fw(5.8)1977 3864 y SDict begin H.R end 1977 3864 a 1977 3929 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.8) cvn H.B /ANN pdfmark end 1977 3929 a Fw(\).)51 b(W)-8 b(e)35 b(asso)s(ciate)g(to)g(a)g(QP-represen)m(tation)94 4059 y Fr(M)c Fw(=)g(\()p Fu(A;)17 b(S;)g(M)5 b(;)17 b(V)k Fw(\))35 b(the)g(QP-represen)m(tation)k Fm(e)-59 b Fu(\026)1948 4074 y Fq(k)1990 4059 y Fw(\()p Fr(M)p Fw(\))30 b(=)h(\()2384 4034 y Fm(e)2361 4059 y Fu(A;)2491 4034 y Fm(e)2478 4059 y Fu(S)6 b(;)p 2588 3979 105 4 v 17 w(M)k(;)p 2736 3979 79 4 v 17 w(V)22 b Fw(\))34 b(de\014ned)i(as)f(follo)m(ws.)94 4175 y(First,)d(w)m(e)i(set)94 4332 y(\(10.6\))1243 4332 y SDict begin H.S end 1243 4332 a 1243 4332 a SDict begin 14 H.A end 1243 4332 a 1243 4332 a SDict begin [ /View [/XYZ H.V] /Dest (equation.10.6) cvn H.B /DEST pdfmark end 1243 4332 a 1243 4252 105 4 v Fu(M)1348 4347 y Fq(i)1404 4332 y Fw(=)27 b Fu(M)1601 4347 y Fq(i)1630 4332 y Fu(;)p 1771 4252 79 4 v 114 w(V)1849 4347 y Fq(i)1905 4332 y Fw(=)h Fu(V)2066 4347 y Fq(i)2191 4332 y Fw(\()p Fu(i)g Fr(6)p Fw(=)g Fu(k)s Fw(\))p Fu(:)94 4494 y Fw(W)-8 b(e)33 b(de\014ne)p 544 4414 105 4 v 34 w Fu(M)649 4509 y Fq(k)724 4494 y Fw(and)p 914 4414 79 4 v 33 w Fu(V)992 4509 y Fq(k)1068 4494 y Fw(b)m(y)94 4699 y(\(10.7\))684 4699 y SDict begin H.S end 684 4699 a 684 4699 a SDict begin 14 H.A end 684 4699 a 684 4699 a SDict begin [ /View [/XYZ H.V] /Dest (equation.10.7) cvn H.B /DEST pdfmark end 684 4699 a 684 4619 105 4 v Fu(M)789 4714 y Fq(k)860 4699 y Fw(=)973 4632 y(k)m(er)18 b Fu(\015)p 973 4676 203 4 v 974 4767 a Fw(im)31 b Fu(\014)1208 4699 y Fr(\010)22 b Fw(im)31 b Fu(\015)d Fr(\010)1636 4632 y Fw(k)m(er)18 b Fu(\013)p 1636 4676 210 4 v 1642 4767 a Fw(im)31 b Fu(\015)1878 4699 y Fr(\010)22 b Fu(V)2034 4714 y Fq(k)2077 4699 y Fu(;)p 2218 4619 79 4 v 114 w(V)2297 4714 y Fq(k)2367 4699 y Fw(=)2638 4632 y(k)m(er)17 b Fu(\014)p 2480 4676 522 4 v 2480 4767 a Fw(k)m(er)h Fu(\014)28 b Fr(\\)22 b Fw(im)31 b Fu(\013)3044 4699 y(:)194 4933 y Fw(It)g(remains)f(to)h (de\014ne)h(the)f(action)f(on)p 1649 4853 105 4 v 31 w Fu(M)42 b Fw(of)30 b(all)f(arro)m(ws)i(in)2473 4908 y Fm(e)2450 4933 y Fu(A)g Fw(\(recall)f(that)g(they)i(are)f(giv)m(en)94 5049 y(b)m(y)40 b(\()274 5049 y SDict begin H.S end 274 5049 a Fw(5.6)399 4985 y SDict begin H.R end 399 4985 a 399 5049 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.6) cvn H.B /ANN pdfmark end 399 5049 a Fw(\))e(and)h(\()709 5049 y SDict begin H.S end 709 5049 a Fw(5.7)834 4985 y SDict begin H.R end 834 4985 a 834 5049 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.7) cvn H.B /ANN pdfmark end 834 5049 a Fw(\)\).)62 b(Th)m(us,)42 b(for)c(ev)m(ery)j(suc)m(h)f (arro)m(w)f Fu(c)p Fw(,)h(w)m(e)g(need)g(to)e(de\014ne)i(a)f(linear)e (map)94 5170 y Fu(c)p 136 5142 79 3 v 26 x Fq(M)243 5170 y Fw(:)p 298 5090 105 4 v 28 w Fu(M)403 5186 y Fq(t)p Fp(\()p Fq(c)p Fp(\))545 5170 y Fr(!)p 673 5090 V 28 w Fu(M)778 5186 y Fq(h)p Fp(\()p Fq(c)p Fp(\))908 5170 y Fw(.)194 5286 y(First,)32 b(w)m(e)h(set)1691 5403 y Fu(c)p 1733 5374 79 3 v 26 x Fq(M)1840 5403 y Fw(=)28 b Fu(c)1986 5418 y Fq(M)94 5539 y Fw(for)33 b(ev)m(ery)h(arro)m(w)f Fu(c)f Fw(not)h(inciden)m(t)f(to)g Fu(k)s Fw(,)h(and)1498 5696 y([)p Fu(b)1566 5711 y Fq(q)1605 5696 y Fu(a)1656 5711 y Fq(p)1695 5696 y Fw(])p 1722 5668 V 26 x Fq(M)1829 5696 y Fw(=)28 b(\()p Fu(b)2012 5711 y Fq(q)2050 5696 y Fu(a)2101 5711 y Fq(p)2141 5696 y Fw(\))2179 5728 y Fq(M)p eop end end %%Page: 42 42 TeXDict begin HPSdict begin 42 41 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.42) cvn H.B /DEST pdfmark end 94 218 a Fs(42)492 b(HARM)25 b(DERKSEN,)f(JERZY)i(WEYMAN,)f(AND)f(ANDREI)g(ZELEVINSKY)94 417 y Fw(for)33 b(all)d Fu(p)i Fw(and)h Fu(q)t Fw(.)194 533 y(T)-8 b(o)29 b(de\014ne)i(the)e(action)g(of)f(the)i(remaining)d (arro)m(ws)j Fu(a)2151 497 y Fq(?)2151 558 y(p)2220 533 y Fw(and)f Fu(b)2447 497 y Fq(?)2447 558 y(q)2487 533 y Fw(,)h(w)m(e)g(assem)m(ble)g(them)e(in)m(to)h(the)94 649 y(op)s(erators)p 1423 734 63 4 v 1423 788 a Fu(\013)f Fw(=)1617 708 y Fm(\000)1663 786 y Fu(b)1704 751 y Fq(?)1704 811 y Fp(1)1827 786 y Fu(b)1868 751 y Fq(?)1868 811 y Fp(2)1990 786 y Fr(\001)17 b(\001)g(\001)98 b Fu(b)2247 751 y Fq(?)2247 811 y(t)2287 708 y Fm(\001)94 948 y Fw(and)p 1650 1177 61 4 v 1650 1258 a Fu(\014)33 b Fw(=)1841 998 y Fm(0)1841 1174 y(B)1841 1233 y(B)1841 1297 y(@)1929 1060 y Fu(a)1980 1024 y Fq(?)1980 1085 y Fp(1)1929 1176 y Fu(a)1980 1140 y Fq(?)1980 1201 y Fp(2)1960 1271 y Fw(.)1960 1304 y(.)1960 1337 y(.)1929 1453 y Fu(a)1980 1417 y Fq(?)1980 1478 y(s)2019 998 y Fm(1)2019 1174 y(C)2019 1233 y(C)2019 1297 y(A)94 1614 y Fw(in)f(the)h(same)g(w)m(a)m(y)h(as)e (in)g(\()1090 1614 y SDict begin H.S end 1090 1614 a Fw(10.3)1263 1550 y SDict begin H.R end 1263 1550 a 1263 1614 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.3) cvn H.B /ANN pdfmark end 1263 1614 a Fw(\).)44 b(Th)m(us,)34 b(w)m(e)g(need)f(to)g(de\014ne)g(linear)e(maps)p 848 1745 63 4 v 848 1800 a Fu(\013)d Fw(:)g Fu(M)1087 1815 y Fp(out)1221 1800 y Fw(=)p 1325 1720 105 4 v 28 w Fu(M)1429 1815 y Fp(in)1520 1800 y Fr(!)p 1647 1720 V 27 w Fu(M)1752 1815 y Fq(k)1795 1800 y Fu(;)p 1936 1719 61 4 v 114 w(\014)33 b Fw(:)p 2079 1720 105 4 v 28 w Fu(M)2184 1815 y Fq(k)2254 1800 y Fr(!)p 2382 1720 V 28 w Fu(M)2486 1815 y Fp(out)2620 1800 y Fw(=)28 b Fu(M)2818 1815 y Fp(in)2881 1800 y Fu(:)94 1979 y Fw(W)-8 b(e)37 b(will)c(use)k(the)f(follo)m(wing)d(notational)g (con)m(v)m(en)m(tion:)51 b(whenev)m(er)38 b(w)m(e)f(ha)m(v)m(e)g(a)f (pair)e Fu(U)3372 1994 y Fp(1)3445 1979 y Fr(\022)g Fu(U)3622 1994 y Fp(2)94 2096 y Fw(of)h(v)m(ector)h(spaces,)i(denote)e(b)m(y)g Fu(\023)c Fw(:)h Fu(U)1480 2111 y Fp(1)1551 2096 y Fr(!)f Fu(U)1749 2111 y Fp(2)1824 2096 y Fw(the)k(inclusion)d(map,)i(and)h(b)m (y)g Fu(\031)g Fw(:)c Fu(U)3198 2111 y Fp(2)3270 2096 y Fr(!)f Fu(U)3467 2111 y Fp(2)3507 2096 y Fu(=U)3622 2111 y Fp(1)94 2212 y Fw(the)i(natural)f(pro)5 b(jection.)43 b(W)-8 b(e)33 b(no)m(w)g(in)m(tro)s(duce)g(the)g(follo)m(wing)d Fv(splitting)k(data)7 b Fw(:)486 2392 y(Cho)s(ose)34 b(a)e(linear)f(map)h Fu(\032)c Fw(:)f Fu(M)1621 2407 y Fp(out)1756 2392 y Fr(!)g Fw(k)m(er)18 b Fu(\015)37 b Fw(suc)m(h)d(that)e Fu(\032\023)d Fw(=)e(id)2847 2407 y Fp(k)n(er)12 b Fq(\015)2997 2392 y Fw(.)94 2392 y SDict begin H.S end 94 2392 a 94 2392 a SDict begin 17.5 H.A end 94 2392 a 94 2392 a SDict begin [ /View [/XYZ H.V] /Dest (equation.10.8) cvn H.B /DEST pdfmark end 94 2392 a Fw(\(10.8\))486 2543 y(Cho)s(ose)34 b(a)e(linear)f(map)h Fu(\033)f Fw(:)d(k)m(er)18 b Fu(\013)q(=)p Fw(im)30 b Fu(\015)j Fr(!)27 b Fw(k)m(er)18 b Fu(\013)33 b Fw(suc)m(h)h(that)e Fu(\031)t(\033)g Fw(=)c(id)3150 2558 y Fp(k)n(er)12 b Fq(\013=)p Fp(im)31 b Fq(\015)3492 2543 y Fw(.)94 2543 y SDict begin H.S end 94 2543 a 94 2543 a SDict begin 17.5 H.A end 94 2543 a 94 2543 a SDict begin [ /View [/XYZ H.V] /Dest (equation.10.9) cvn H.B /DEST pdfmark end 94 2543 a Fw(\(10.9\))94 2723 y(Then)j(w)m(e)g(de\014ne:)94 3058 y(\(10.10\))1121 3058 y SDict begin H.S end 1121 3058 a 1121 3058 a SDict begin 14 H.A end 1121 3058 a 1121 3058 a SDict begin [ /View [/XYZ H.V] /Dest (equation.10.10) cvn H.B /DEST pdfmark end 1121 3058 a 1121 3003 63 4 v Fu(\013)28 b Fw(=)1315 2798 y Fm(0)1315 2974 y(B)1315 3033 y(B)1315 3097 y(@)1402 2882 y Fr(\000)p Fu(\031)t(\032)1429 2998 y Fr(\000)p Fu(\015)1471 3115 y Fw(0)1471 3231 y(0)1589 2798 y Fm(1)1589 2974 y(C)1589 3033 y(C)1589 3097 y(A)1692 3058 y Fu(;)p 1834 2977 61 4 v 115 w(\014)33 b Fw(=)2025 2978 y Fm(\000)2071 3056 y Fw(0)83 b Fu(\023)g(\023\033)88 b Fw(0)2546 2978 y Fm(\001)2608 3058 y Fu(:)94 3336 y SDict begin H.S end 94 3336 a 94 3336 a SDict begin 14 H.A end 94 3336 a 94 3336 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.10.7) cvn H.B /DEST pdfmark end 94 3336 a 104 x FA(Prop)s(osition)33 b(10.7.)40 b Fv(The)31 b(ab)-5 b(ove)32 b(de\014nitions)f(make)36 b Fm(e)-59 b Fu(\026)2209 3455 y Fq(k)2251 3440 y Fw(\()p Fr(M)p Fw(\))27 b(=)h(\()p 2616 3360 105 4 v Fu(M)10 b(;)p 2764 3360 79 4 v 17 w(V)22 b Fw(\))32 b Fv(a)g(de)-5 b(c)g(or)g(ate)g(d)32 b(r)-5 b(epr)g(e-)94 3570 y(sentation)35 b(of)f Fw(\()693 3545 y Fm(e)670 3570 y Fu(A)q(;)801 3545 y Fm(e)788 3570 y Fu(S)5 b Fw(\))p Fv(.)94 3772 y(Pr)-5 b(o)g(of.)42 b Fw(The)32 b(prop)s(ert)m(y)h(that)e(some)h(p)s(o)m(w)m(er)g(of)f Fl(m)p Fw(\()1974 3746 y Fm(e)1951 3772 y Fu(A)q Fw(\))g(annihilates)p 2587 3692 105 4 v 30 w Fu(M)42 b Fw(is)32 b(immediate)d(from)h(the)94 3888 y(de\014nitions)i(and)f(the)h(corresp)s(onding)g(prop)s(ert)m(y)g (for)g Fu(M)10 b Fw(.)43 b(Th)m(us,)34 b(w)m(e)f(only)e(need)h(to)g (sho)m(w)h(that)94 4013 y(\()p Fu(@)183 4028 y Fq(c)232 3987 y Fm(e)219 4013 y Fu(S)5 b Fw(\))p 322 3984 79 3 v 26 x Fq(M)434 4013 y Fw(=)33 b(0)j(for)f(ev)m(ery)i(arro)m(w)f Fu(c)g Fw(in)1534 3987 y Fm(e)1512 4013 y Fu(A)p Fw(.)53 b(If)36 b Fu(c)f Fw(is)g(not)h(inciden)m(t)f(to)h Fu(k)s Fw(,)g(the)h(desired)f(statemen)m(t)94 4129 y(follo)m(ws)d(from)f(\() 685 4129 y SDict begin H.S end 685 4129 a Fw(6.6)810 4065 y SDict begin H.R end 810 4065 a 810 4129 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.6.6) cvn H.B /ANN pdfmark end 810 4129 a Fw(\).)47 b(If)34 b Fu(c)f Fw(is)g(one)i(of)e(the)h (arro)m(ws)g([)p Fu(b)2038 4144 y Fq(q)2077 4129 y Fu(a)2128 4144 y Fq(p)2168 4129 y Fw(],)g(then,)h(in)e(view)h(of)f(\()2998 4129 y SDict begin H.S end 2998 4129 a Fw(6.5)3123 4065 y SDict begin H.R end 3123 4065 a 3123 4129 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.6.5) cvn H.B /ANN pdfmark end 3123 4129 a Fw(\))g(and)h(\()3423 4129 y SDict begin H.S end 3423 4129 a Fw(10.4)3596 4065 y SDict begin H.R end 3596 4065 a 3596 4129 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.4) cvn H.B /ANN pdfmark end 3596 4129 a Fw(\),)94 4245 y(it)e(is)g(enough)h(to)f(sho)m(w)i(that)1570 4384 y Fu(a)1621 4343 y Fq(?)1621 4409 y(p)1661 4384 y Fu(b)1702 4343 y Fq(?)1702 4409 y(q)1764 4384 y Fw(+)22 b Fu(\015)1913 4399 y Fq(p;q)2033 4384 y Fw(=)28 b(0)94 4562 y(for)34 b(all)d Fu(p)j Fw(and)g Fu(q)t Fw(.)47 b(In)34 b(other)g(w)m(ords,)h(w)m(e)g(need)g(to)f(sho)m(w)h(that)p 2413 4481 61 4 v 33 w Fu(\014)p 2474 4507 63 4 v 6 w(\013)30 b Fw(=)g Fr(\000)p Fu(\015)5 b Fw(;)35 b(But)f(this)f(follo)m(ws)g(at) 94 4678 y(once)h(b)m(y)f(m)m(ultiplying)c(the)k(ro)m(w)g(and)g(column)f (giv)m(en)g(b)m(y)i(\()2287 4678 y SDict begin H.S end 2287 4678 a Fw(10.10)2509 4614 y SDict begin H.R end 2509 4614 a 2509 4678 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.10) cvn H.B /ANN pdfmark end 2509 4678 a Fw(\).)194 4809 y(It)k(remains)e(to)h(sho)m(w)h(that)g(\()p Fu(@)1349 4824 y Fq(a)1386 4805 y Fg(?)1386 4840 y(p)1440 4783 y Fm(e)1427 4809 y Fu(S)6 b Fw(\))p 1531 4780 79 3 v 26 x Fq(M)1645 4809 y Fw(=)36 b(0)h(and)h(\()p Fu(@)2127 4824 y Fq(b)2157 4805 y Fg(?)2157 4840 y(q)2211 4783 y Fm(e)2197 4809 y Fu(S)6 b Fw(\))p 2301 4780 V 26 x Fq(M)2416 4809 y Fw(=)36 b(0)h(for)f(all)g Fu(p)h Fw(and)g Fu(q)t Fw(.)58 b(W)-8 b(e)38 b(\014rst)94 4933 y(deal)32 b(with)h(the)g(\014rst)g(equalit)m(y)-8 b(.)43 b(Remem)m(b)s(ering)31 b(\()1950 4933 y SDict begin H.S end 1950 4933 a Fw(5.8)2074 4869 y SDict begin H.R end 2074 4869 a 2074 4933 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.8) cvn H.B /ANN pdfmark end 2074 4933 a Fw(\))i(and)g(\()2373 4933 y SDict begin H.S end 2373 4933 a Fw(5.9)2497 4869 y SDict begin H.R end 2497 4869 a 2497 4933 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.9) cvn H.B /ANN pdfmark end 2497 4933 a Fw(\),)g(w)m(e)g(see)h(that)780 5134 y(\()p Fu(@)869 5149 y Fq(a)906 5130 y Fg(?)906 5166 y(p)960 5109 y Fm(e)947 5134 y Fu(S)6 b Fw(\))p 1051 5106 79 3 v 26 x Fq(M)1158 5134 y Fw(=)27 b(\()1299 5039 y Fm(X)1354 5249 y Fq(q)1460 5134 y Fu(b)1501 5093 y Fq(?)1501 5159 y(q)1540 5134 y Fw([)p Fu(b)1608 5149 y Fq(q)1647 5134 y Fu(a)1698 5149 y Fq(p)1738 5134 y Fw(]\))p 1803 5106 V 26 x Fq(M)1910 5134 y Fw(=)g(\()2051 5039 y Fm(X)2106 5249 y Fq(q)2195 5134 y Fw(\()p Fu(b)2274 5093 y Fq(?)2274 5159 y(q)2314 5134 y Fw(\))p 2352 5106 V 26 x Fq(M)2431 5134 y Fw(\()p Fu(b)2510 5149 y Fq(q)2548 5134 y Fw(\))2586 5149 y Fq(M)2665 5134 y Fw(\)\()p Fu(a)2792 5149 y Fq(p)2832 5134 y Fw(\))2870 5149 y Fq(M)2949 5134 y Fu(:)94 5412 y Fw(Th)m(us)35 b(it)c(su\016ces)k(to)d(sho)m(w)i(that) 1467 5497 y Fm(X)1522 5707 y Fq(q)1611 5592 y Fw(\()p Fu(b)1690 5551 y Fq(?)1690 5617 y(q)1730 5592 y Fw(\))p 1768 5564 V 26 x Fq(M)1847 5592 y Fw(\()p Fu(b)1926 5607 y Fq(q)1965 5592 y Fw(\))2003 5607 y Fq(M)2109 5592 y Fw(=)28 b(0)p Fu(;)p eop end end %%Page: 43 43 TeXDict begin HPSdict begin 43 42 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.43) cvn H.B /DEST pdfmark end 94 218 a 1179 w Fs(QUIVERS)31 b(WITH)i(POTENTIALS)g(I)1102 b(43)94 417 y Fw(or)33 b(equiv)-5 b(alen)m(tly)d(,)p 775 362 63 4 v 32 w Fu(\013\014)34 b Fw(=)27 b(0.)43 b(In)33 b(view)g(of)f(\()1644 417 y SDict begin H.S end 1644 417 a Fw(10.10)1866 353 y SDict begin H.R end 1866 353 a 1866 417 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.10) cvn H.B /ANN pdfmark end 1866 417 a Fw(\),)h(w)m(e)h(ha)m(v)m(e)p 1436 719 63 4 v 1436 774 a Fu(\013)q(\014)f Fw(=)1691 514 y Fm(0)1691 689 y(B)1691 749 y(B)1691 812 y(@)1778 598 y Fr(\000)p Fu(\031)t(\032\014)1805 714 y Fr(\000)p Fu(\015)5 b(\014)1877 830 y Fw(0)1877 946 y(0)2025 514 y Fm(1)2025 689 y(C)2025 749 y(C)2025 812 y(A)2140 774 y Fw(=)28 b(0)p Fu(;)94 1130 y Fw(as)33 b(desired)g(\(the)g(equalit)m (y)f Fu(\031)t(\032\014)h Fw(=)28 b(0)k(is)g(immediate)d(from)i(the)i (de\014nitions,)f(while)g Fu(\015)5 b(\014)33 b Fw(=)27 b(0)32 b(b)m(y)94 1246 y(Lemma)443 1246 y SDict begin H.S end 443 1246 a Fw(10.6)616 1182 y SDict begin H.R end 616 1182 a 616 1246 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.10.6) cvn H.B /ANN pdfmark end 616 1246 a Fw(\).)194 1377 y(The)i(remaining)c(equalit)m(y)i(\()p Fu(@)1310 1392 y Fq(b)1340 1373 y Fg(?)1340 1408 y(\027)1397 1352 y Fm(e)1383 1377 y Fu(S)6 b Fw(\))p 1487 1348 79 3 v 26 x Fq(M)1594 1377 y Fw(=)27 b(0)33 b(is)f(pro)m(v)m(ed)i(in)e(a)g (similar)d(w)m(a)m(y)-8 b(.)45 b(W)-8 b(e)33 b(ha)m(v)m(e)812 1581 y(\()p Fu(@)901 1596 y Fq(b)931 1577 y Fg(?)931 1612 y(q)985 1556 y Fm(e)971 1581 y Fu(S)6 b Fw(\))p 1075 1552 V 26 x Fq(M)1182 1581 y Fw(=)27 b(\()1323 1486 y Fm(X)1377 1695 y Fq(p)1467 1581 y Fw([)p Fu(b)1535 1596 y Fq(q)1574 1581 y Fu(a)1625 1596 y Fq(p)1665 1581 y Fw(])p Fu(a)1743 1540 y Fq(?)1743 1605 y(p)1783 1581 y Fw(\))p 1821 1552 V 26 x Fq(M)1927 1581 y Fw(=)h(\()p Fu(b)2110 1596 y Fq(q)2148 1581 y Fw(\))2186 1596 y Fq(M)2282 1486 y Fm(X)2336 1695 y Fq(p)2426 1581 y Fw(\()p Fu(a)2515 1596 y Fq(p)2554 1581 y Fw(\))2592 1596 y Fq(M)2671 1581 y Fw(\()p Fu(a)2760 1540 y Fq(?)2760 1605 y(p)2800 1581 y Fw(\))p 2838 1552 V 26 x Fq(M)2917 1581 y Fu(:)94 1855 y Fw(Th)m(us,)35 b(it)c(su\016ces)k(to)d(observ)m(e)j(that)1302 2031 y Fu(\013)p 1365 1950 61 4 v 1 w(\014)d Fw(=)1556 1951 y Fm(\000)1602 2029 y Fw(0)82 b Fu(\013)q(\023)i(\013)q(\023\033)j Fw(0)2202 1951 y Fm(\001)2275 2031 y Fw(=)28 b(0)p Fu(:)3584 2207 y Fh(\003)94 2280 y SDict begin H.S end 94 2280 a 94 2280 a SDict begin 14 H.A end 94 2280 a 94 2280 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.10.8) cvn H.B /DEST pdfmark end 94 2280 a 112 x FA(Example)34 b(10.8.)40 b Fw(Let)29 b(us)i(illustrate)c(the)k(de\014nition)d(of)i(the)g(op)s (eration)j Fm(e)-60 b Fu(\026)2889 2407 y Fq(k)2961 2392 y Fw(b)m(y)31 b(a)f(sp)s(ecial)e(case)94 2508 y(where)34 b(the)f(v)m(ertex)h Fu(k)i Fw(is)c(a)g(sink)g(or)h(a)f(source.)44 b(First)32 b(supp)s(ose)i Fu(k)h Fw(is)d(a)g(sink,)h(that)f(is,)g (there)h(are)94 2624 y(no)h(arro)m(ws)h Fu(b)g Fw(with)e Fu(tb)e Fw(=)f Fu(k)s Fw(.)48 b(Then)35 b(w)m(e)h(ha)m(v)m(e)f Fu(M)1908 2639 y Fp(out)2045 2624 y Fw(=)30 b Fr(f)p Fw(0)p Fr(g)p Fw(,)j(hence)j Fu(\014)g Fw(=)30 b(0)k(and)g Fu(\015)h Fw(=)30 b(0.)48 b(Th)m(us,)94 2741 y(\()132 2741 y SDict begin H.S end 132 2741 a Fw(10.7)306 2676 y SDict begin H.R end 306 2676 a 306 2741 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.7) cvn H.B /ANN pdfmark end 306 2741 a Fw(\))32 b(simpli\014es)f(to)p 1138 2801 105 4 v 1138 2881 a Fu(M)1243 2896 y Fq(k)1313 2881 y Fw(=)d(k)m(er)17 b Fu(\013)23 b Fr(\010)g Fu(V)1805 2896 y Fq(k)1847 2881 y Fu(;)p 1989 2801 79 4 v 115 w(V)2067 2896 y Fq(k)2137 2881 y Fw(=)28 b(cok)m(er)34 b Fu(\013)f(:)94 3044 y Fw(The)39 b(arro)m(w)e(span)834 3019 y Fm(e)812 3044 y Fu(A)g Fw(is)g(obtained)g(from)f Fu(A)i Fw(b)m(y)g(rev)m(ersing) g(all)e(arro)m(ws)i Fu(a)f Fw(with)h Fu(ha)e Fw(=)g Fu(k)s Fw(,)j(that)94 3169 y(is,)44 b(replacing)c(ev)m(ery)j(suc)m(h)g(arro)m (w)f Fu(a)f Fw(with)g Fu(a)1809 3133 y Fq(?)1849 3169 y Fw(.)70 b(Th)m(us,)45 b Fu(k)g Fw(b)s(ecomes)d(a)f(source)h(for)3302 3144 y Fm(e)3279 3169 y Fu(A)p Fw(,)i(hence)p 94 3205 105 4 v 94 3285 a Fu(M)199 3300 y Fq(in)320 3285 y Fw(=)50 b Fr(f)p Fw(0)p Fr(g)45 b Fw(and)p 843 3231 63 4 v 46 w Fu(\013)50 b Fw(=)g(0.)82 b(The)47 b(c)m(hoice)f(of)f(splitting)e (data)i(\()2551 3285 y SDict begin H.S end 2551 3285 a Fw(10.8)2725 3221 y SDict begin H.R end 2725 3221 a 2725 3285 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.8) cvn H.B /ANN pdfmark end 2725 3285 a Fw(\))g(and)h(\()3049 3285 y SDict begin H.S end 3049 3285 a Fw(10.9)3222 3221 y SDict begin H.R end 3222 3221 a 3222 3285 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.9) cvn H.B /ANN pdfmark end 3222 3285 a Fw(\))g(b)s(ecomes)94 3402 y(immaterial,)29 b(and)j(the)h(second)h(equalit)m(y)f(in)e(\()1810 3402 y SDict begin H.S end 1810 3402 a Fw(10.10)2032 3337 y SDict begin H.R end 2032 3337 a 2032 3402 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.10) cvn H.B /ANN pdfmark end 2032 3402 a Fw(\))i(simpli\014es)e(to)p 1631 3502 61 4 v 1631 3583 a Fu(\014)j Fw(=)1823 3503 y Fm(\000)1869 3581 y Fu(\023)83 b Fw(0)2035 3503 y Fm(\001)2097 3583 y Fu(:)94 3783 y Fw(Note)25 b(that)f(w)m(e)h(ha)m(v)m(e)898 3758 y Fm(e)876 3783 y Fu(A)951 3795 y Fp(^)949 3812 y Fq(k)r(;)1010 3795 y Fp(^)1008 3812 y Fq(k)1078 3783 y Fw(=)i Fu(A)1256 3795 y Fp(^)1254 3812 y Fq(k)r(;)1315 3795 y Fp(^)1313 3812 y Fq(k)1355 3783 y Fw(,)f(and)e(the)h(p)s(oten)m (tial)2165 3758 y Fm(e)2152 3783 y Fu(S)33 b Fr(2)28 b Fu(R)q Fr(hh)2514 3758 y Fm(e)2492 3783 y Fu(A)2567 3795 y Fp(^)2565 3812 y Fq(k)r(;)2626 3795 y Fp(^)2624 3812 y Fq(k)2666 3783 y Fr(ii)23 b Fw(is)h(naturally)f(iden)m(ti\014ed) 94 3903 y(with)33 b Fu(S)6 b Fw(.)194 4019 y(The)28 b(case)f(where)h Fu(k)i Fw(is)c(a)h(source)g(is)g(completely)e(similar.)39 b(In)27 b(this)f(case)i(w)m(e)f(ha)m(v)m(e)h Fu(M)3291 4034 y Fp(in)3382 4019 y Fw(=)g Fr(f)p Fw(0)p Fr(g)p Fw(,)94 4135 y(hence)34 b Fu(\013)29 b Fw(=)e(0)33 b(and)f Fu(\015)h Fw(=)28 b(0.)43 b(It)32 b(follo)m(ws)g(that)p 1140 4231 105 4 v 1140 4311 a Fu(M)1245 4326 y Fq(k)1315 4311 y Fw(=)c(cok)m(er)33 b Fu(\014)28 b Fr(\010)23 b Fu(V)1913 4326 y Fq(k)1955 4311 y Fu(;)p 2097 4231 79 4 v 115 w(V)2175 4326 y Fq(k)2245 4311 y Fw(=)28 b(k)m(er)18 b Fu(\014)38 b(;)94 4487 y Fw(and)33 b(the)g(map)p 669 4432 63 4 v 32 w Fu(\013)28 b Fw(:)p 814 4407 105 4 v 28 w Fu(M)919 4502 y Fp(in)1009 4487 y Fw(=)g Fu(M)1207 4502 y Fp(out)1341 4487 y Fr(!)p 1468 4407 V 27 w Fu(M)1573 4502 y Fq(k)1648 4487 y Fw(is)33 b(giv)m(en)f(b)m(y)p 1618 4663 63 4 v 1618 4718 a Fu(\013)c Fw(=)1812 4578 y Fm(\022)1885 4658 y Fr(\000)p Fu(\031)1929 4775 y Fw(0)2021 4578 y Fm(\023)2111 4718 y Fu(:)194 4952 y Fw(In)39 b(b)s(oth)f(cases,) 45 b Fm(e)-59 b Fu(\026)896 4967 y Fq(k)976 4952 y Fw(coincides)39 b(with)e(the)i(\\extended)h(re\015ection)f(functor")f(in)m(tro)s(duced) g(in)94 5069 y([)121 5069 y SDict begin H.S end 121 5069 a Fw(28)219 5005 y SDict begin H.R end 219 5005 a 219 5069 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.mrz) cvn H.B /ANN pdfmark end 219 5069 a Fw(];)44 b(furthermore,)d(if)e(w)m (e)h(ignore)f(the)h(decorations,)i(it)d(b)s(ecomes)h(the)g(classical)e (Bernstein-)94 5185 y(Gelfand-P)m(onomarev)32 b(re\015ection)g(functor) h(at)f Fu(k)s Fw(,)h(see)h([)2151 5185 y SDict begin H.S end 2151 5185 a Fw(3)2200 5121 y SDict begin H.R end 2200 5121 a 2200 5185 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.bgp) cvn H.B /ANN pdfmark end 2200 5185 a Fw(].)194 5370 y(No)m(w)f(w)m(e)h(return)f(to)f(the)h(case)h(of)e(an)g (arbitrary)g(v)m(ertex)i Fu(k)s Fw(.)94 5389 y SDict begin H.S end 94 5389 a 94 5389 a SDict begin 14 H.A end 94 5389 a 94 5389 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.10.9) cvn H.B /DEST pdfmark end 94 5389 a 166 x FA(Prop)s(osition)43 b(10.9.)i Fv(The)c(isomorphism)d(class)i(of)h(the)g(de)-5 b(c)g(or)g(ate)g(d)40 b(r)-5 b(epr)g(esentation)44 b Fm(e)-59 b Fu(\026)3424 5570 y Fq(k)3466 5555 y Fw(\()p Fr(M)p Fw(\))94 5671 y Fv(do)-5 b(es)35 b(not)g(dep)-5 b(end)33 b(on)i(the)g(choic)-5 b(e)34 b(of)g(the)h(splitting)g(data)f Fw(\()2304 5671 y SDict begin H.S end 2304 5671 a Fw(10.8)2478 5607 y SDict begin H.R end 2478 5607 a 2478 5671 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.8) cvn H.B /ANN pdfmark end 2478 5671 a Fw(\))h Fv({)f Fw(\()2673 5671 y SDict begin H.S end 2673 5671 a Fw(10.9)2847 5607 y SDict begin H.R end 2847 5607 a 2847 5671 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.9) cvn H.B /ANN pdfmark end 2847 5671 a Fw(\))p Fv(.)p eop end end %%Page: 44 44 TeXDict begin HPSdict begin 44 43 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.44) cvn H.B /DEST pdfmark end 94 218 a Fs(44)492 b(HARM)25 b(DERKSEN,)f(JERZY)i(WEYMAN,)f(AND)f(ANDREI)g(ZELEVINSKY)94 417 y Fv(Pr)-5 b(o)g(of.)42 b Fw(W)-8 b(e)28 b(ha)m(v)m(e)h(the)f (follo)m(wing)d(freedom)j(in)f(c)m(ho)s(osing)g(the)i(splitting)c (data:)41 b(one)28 b(can)g(replace)94 533 y Fu(\032)g Fw(:)g Fu(M)321 548 y Fp(out)455 533 y Fr(!)g Fw(k)m(er)17 b Fu(\015)38 b Fw(with)32 b Fu(\032)1090 497 y Fn(0)1141 533 y Fw(=)27 b Fu(\032)c Fw(+)e Fu(\030)5 b(\015)37 b Fw(for)32 b(some)g(linear)f(map)g Fu(\030)h Fw(:)c(im)j Fu(\015)h Fr(!)c Fw(k)m(er)17 b Fu(\015)5 b Fw(,)33 b(and)f(replace)94 649 y Fu(\033)39 b Fw(:)34 b(k)m(er)18 b Fu(\013)q(=)p Fw(im)34 b Fu(\015)40 b Fr(!)34 b Fw(k)m(er)17 b Fu(\013)38 b Fw(b)m(y)f Fu(\033)1321 613 y Fn(0)1379 649 y Fw(=)d Fu(\033)29 b Fw(+)24 b Fu(\021)41 b Fw(for)36 b(some)g(linear)f(map)h Fu(\021)i Fw(:)c(k)m(er)18 b Fu(\013)q(=)p Fw(im)34 b Fu(\015)39 b Fr(!)34 b Fw(im)h Fu(\015)5 b Fw(.)94 782 y(Let)p 267 728 63 4 v 31 w Fu(\013)330 746 y Fn(0)383 782 y Fw(and)p 571 701 61 4 v 31 w Fu(\014)632 720 y Fn(0)685 782 y Fw(b)s(e)31 b(the)g(maps)f(obtained)g(b)m(y)h(replacing) e Fu(\032)i Fw(with)f Fu(\032)2533 746 y Fn(0)2557 782 y Fw(,)h(and)f Fu(\033)35 b Fw(with)30 b Fu(\033)3171 746 y Fn(0)3225 782 y Fw(in)f(\()3374 782 y SDict begin H.S end 3374 782 a Fw(10.10)3596 718 y SDict begin H.R end 3596 718 a 3596 782 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.10) cvn H.B /ANN pdfmark end 3596 782 a Fw(\).)94 915 y(It)46 b(is)e(enough)h(to)g(sho)m(w)h(that)f Fu( )p 1348 860 63 4 v 4 w(\013)k Fw(=)p 1584 860 V 49 w Fu(\013)1647 879 y Fn(0)1715 915 y Fw(and)p 1917 834 61 4 v 45 w Fu(\014)1978 852 y Fn(0)2001 915 y Fu( )k Fw(=)p 2241 834 V 48 w Fu(\014)e Fw(for)44 b(some)h(linear)e (automorphism)94 1032 y Fu( )c Fw(:)p 258 952 105 4 v 35 w Fu(M)363 1047 y Fq(k)440 1032 y Fr(!)p 575 952 V 35 w Fu(M)680 1047 y Fq(k)722 1032 y Fw(.)56 b(Decomp)s(osing)p 1407 952 V 35 w Fu(M)1512 1047 y Fq(k)1592 1032 y Fw(as)37 b(in)f(\()1872 1032 y SDict begin H.S end 1872 1032 a Fw(10.7)2045 967 y SDict begin H.R end 2045 967 a 2045 1032 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.7) cvn H.B /ANN pdfmark end 2045 1032 a Fw(\),)i(w)m(e)f(de\014ne)h Fu( )j Fw(as)c(the)g(blo)s(c)m(k-triangular)94 1148 y(matrix)1377 1431 y Fu( )31 b Fw(=)1575 1171 y Fm(0)1575 1346 y(B)1575 1406 y(B)1575 1470 y(@)1662 1255 y Fu(I)91 b(\031)t(\030)127 b Fw(0)e(0)1663 1371 y(0)112 b Fu(I)118 b Fr(\000)p Fu(\021)89 b Fw(0)1663 1487 y(0)113 b(0)150 b Fu(I)132 b Fw(0)1663 1603 y(0)113 b(0)151 b(0)124 b Fu(I)2248 1171 y Fm(1)2248 1346 y(C)2248 1406 y(C)2248 1470 y(A)2352 1431 y Fu(;)94 1753 y Fw(where)44 b Fu(I)50 b Fw(stands)43 b(for)e(the)i(iden)m(tit)m (y)e(transformation.)70 b(The)43 b(in)m(v)m(ertibilit)m(y)d(of)i Fu( )j Fw(is)d(ob)m(vious,)94 1881 y(and)49 b(the)g(desired)g (equalities)f Fu( )p 1346 1826 63 4 v 4 w(\013)56 b Fw(=)p 1594 1826 V 54 w Fu(\013)1657 1844 y Fn(0)1729 1881 y Fw(and)p 1935 1800 61 4 v 49 w Fu(\014)1995 1818 y Fn(0)2019 1881 y Fu( )j Fw(=)p 2271 1800 V 54 w Fu(\014)c Fw(are)48 b(c)m(hec)m(k)m(ed)k(b)m(y)d(direct)g(matrix)94 1997 y(m)m(ultiplication.)2865 b Fh(\003)94 2079 y SDict begin H.S end 94 2079 a 94 2079 a SDict begin 14 H.A end 94 2079 a 94 2079 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.10.10) cvn H.B /DEST pdfmark end 94 2079 a 98 x FA(Prop)s(osition)34 b(10.10.)41 b Fv(The)32 b(right-e)-5 b(quivalenc)g(e)32 b(class)h(of)g(the)g(r)-5 b(epr)g(esentation)37 b Fm(e)-59 b Fu(\026)3158 2192 y Fq(k)3200 2177 y Fw(\()p Fr(M)p Fw(\))33 b Fv(is)g(de-)94 2294 y(termine)-5 b(d)35 b(by)g(the)g (right-e)-5 b(quivalenc)g(e)33 b(class)h(of)h Fr(M)p Fv(.)94 2474 y(Pr)-5 b(o)g(of.)42 b Fw(Let)31 b Fu(')h Fw(b)s(e)g(an)f(automorphism)e(of)i Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fw(,)g(and)h(let)f Fr(M)2493 2438 y Fn(0)2544 2474 y Fw(=)c(\()p Fu(A;)17 b(')p Fw(\()p Fu(S)6 b Fw(\))p Fu(;)17 b(M)3156 2438 y Fn(0)3179 2474 y Fu(;)g(V)3301 2438 y Fn(0)3325 2474 y Fw(\))31 b(b)s(e)h(the)94 2590 y(QP-represen)m(tation)g(de\014ned)g(as)f(follo)m(ws:)41 b Fu(V)1793 2554 y Fn(0)1844 2590 y Fw(=)27 b Fu(V)53 b Fw(and)30 b Fu(M)2348 2554 y Fn(0)2400 2590 y Fw(=)e Fu(M)41 b Fw(as)31 b Fu(R)q Fw(-mo)s(dules,)f(while)g(the)94 2706 y Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fw(-actions)h(in)h Fu(M)43 b Fw(and)33 b Fu(M)1306 2670 y Fn(0)1363 2706 y Fw(are)f(related)g(b)m(y)94 2874 y(\(10.11\))1270 2874 y SDict begin H.S end 1270 2874 a 1270 2874 a SDict begin 14 H.A end 1270 2874 a 1270 2874 a SDict begin [ /View [/XYZ H.V] /Dest (equation.10.11) cvn H.B /DEST pdfmark end 1270 2874 a Fu(u)1326 2889 y Fq(M)1432 2874 y Fw(=)c Fu(')p Fw(\()p Fu(u)p Fw(\))1732 2889 y Fq(M)1807 2870 y Ff(0)1930 2874 y Fw(\()p Fu(u)f Fr(2)h Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fw(\))94 3041 y(\(note)39 b(that)g(\()611 3041 y SDict begin H.S end 611 3041 a Fw(10.11)833 2977 y SDict begin H.R end 833 2977 a 833 3041 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.11) cvn H.B /ANN pdfmark end 833 3041 a Fw(\))g(indeed)g(de\014nes)h(a)f(represen)m(tation)g(of)f(\()p Fu(A;)17 b(')p Fw(\()p Fu(S)6 b Fw(\)\))38 b(in)g(view)h(of)g(Prop)s (osi-)94 3158 y(tion)303 3158 y SDict begin H.S end 303 3158 a Fw(3.7)427 3093 y SDict begin H.R end 427 3093 a 427 3158 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.3.7) cvn H.B /ANN pdfmark end 427 3158 a Fw(\).)66 b(T)-8 b(o)40 b(pro)m(v)m(e)i(Prop)s(osition)1512 3158 y SDict begin H.S end 1512 3158 a Fw(10.10)1734 3093 y SDict begin H.R end 1734 3093 a 1734 3158 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.10.10) cvn H.B /ANN pdfmark end 1734 3158 a Fw(,)g(it)d(su\016ces)k(to)c(sho)m(w)j(that)e(the)g(represen)m (tations)98 3274 y Fm(e)-59 b Fu(\026)153 3289 y Fq(k)196 3274 y Fw(\()p Fr(M)p Fw(\))32 b(and)37 b Fm(e)-59 b Fu(\026)673 3289 y Fq(k)715 3274 y Fw(\()p Fr(M)873 3238 y Fn(0)896 3274 y Fw(\))32 b(are)h(righ)m(t-equiv)-5 b(alen)m(t.)194 3390 y(W)d(e)44 b(retain)e(all)f(the)j(ab)s(o)m(v)m(e)g (notation)e(related)h(to)g Fr(M)g Fw(and)48 b Fm(e)-60 b Fu(\026)2566 3405 y Fq(k)2608 3390 y Fw(\()p Fr(M)p Fw(\);)49 b(in)42 b(particular,)i Fu(\013)q(;)17 b(\014)94 3506 y Fw(and)36 b Fu(\015)k Fw(stand)c(for)e(the)i(linear)e(maps)h(in) f(the)i(triangle)d(\()2186 3506 y SDict begin H.S end 2186 3506 a Fw(10.5)2360 3442 y SDict begin H.R end 2360 3442 a 2360 3506 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.5) cvn H.B /ANN pdfmark end 2360 3506 a Fw(\).)51 b(Let)35 b Fu(\013)2716 3470 y Fn(0)2739 3506 y Fu(;)17 b(\014)2844 3470 y Fn(0)2902 3506 y Fw(and)35 b Fu(\015)3150 3470 y Fn(0)3209 3506 y Fw(denote)h(the)94 3623 y(corresp)s(onding)26 b(maps)f(for)g(the)h(represen)m(tation)g Fr(M)2015 3586 y Fn(0)2038 3623 y Fw(.)41 b(Our)26 b(\014rst)g(order)g(of)f(business)i (is)e(to)g(relate)94 3739 y(these)34 b(maps)f(to)f Fu(\013)q(;)17 b(\014)37 b Fw(and)c Fu(\015)5 b Fw(.)194 3855 y(W)-8 b(e)33 b(can)g(write)f(the)h(action)f(of)g Fu(')g Fw(on)h(the)g(arro)m (ws)g Fu(a)2125 3870 y Fp(1)2165 3855 y Fu(;)17 b(:)g(:)g(:)e(;)i(a) 2434 3870 y Fq(s)2504 3855 y Fw(as)32 b(follo)m(ws:)94 4023 y(\(10.12\))794 4023 y SDict begin H.S end 794 4023 a 794 4023 a SDict begin 14 H.A end 794 4023 a 794 4023 a SDict begin [ /View [/XYZ H.V] /Dest (equation.10.12) cvn H.B /DEST pdfmark end 794 4023 a 811 3942 a Fm(\000)857 4021 y Fu(')p Fw(\()p Fu(a)1010 4036 y Fp(1)1049 4021 y Fw(\))83 b Fu(')p Fw(\()p Fu(a)1323 4036 y Fp(2)1362 4021 y Fw(\))g Fr(\001)17 b(\001)g(\001)98 b Fu(')p Fw(\()p Fu(a)1852 4036 y Fq(s)1889 4021 y Fw(\))1927 3942 y Fm(\001)2000 4023 y Fw(=)2104 3942 y Fm(\000)2150 4021 y Fu(a)2201 4036 y Fp(1)2323 4021 y Fu(a)2374 4036 y Fp(2)2497 4021 y Fr(\001)17 b(\001)g(\001)98 b Fu(a)2764 4036 y Fq(s)2801 3942 y Fm(\001)2863 4023 y Fu(C)r(;)94 4192 y Fw(where)32 b Fu(C)j Fw(=)27 b Fu(C)652 4207 y Fp(0)709 4192 y Fw(+)17 b Fu(C)872 4207 y Fp(1)941 4192 y Fw(is)30 b(an)g(in)m(v)m(ertible)g Fu(s)17 b Fr(\002)h Fu(s)30 b Fw(matrix)e(as)j(in)e(\()2414 4192 y SDict begin H.S end 2414 4192 a Fw(5.14)2588 4127 y SDict begin H.R end 2588 4127 a 2588 4192 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.14) cvn H.B /ANN pdfmark end 2588 4192 a Fw(\).)42 b(Similarly)-8 b(,)27 b(the)k(action)e(of) 94 4308 y Fu(')k Fw(on)f(the)h(arro)m(ws)h Fu(b)848 4323 y Fp(1)887 4308 y Fu(;)17 b(:)g(:)g(:)f(;)h(b)1147 4323 y Fq(s)1217 4308 y Fw(can)33 b(b)s(e)f(written)h(as)94 4667 y(\(10.13\))1407 4667 y SDict begin H.S end 1407 4667 a 1407 4667 a SDict begin 14 H.A end 1407 4667 a 1407 4667 a SDict begin [ /View [/XYZ H.V] /Dest (equation.10.13) cvn H.B /DEST pdfmark end 1407 4667 a 1423 4406 a Fm(0)1423 4582 y(B)1423 4642 y(B)1423 4705 y(@)1511 4468 y Fu(')p Fw(\()p Fu(b)1654 4483 y Fp(1)1693 4468 y Fw(\))1511 4584 y Fu(')p Fw(\()p Fu(b)1654 4599 y Fp(2)1693 4584 y Fw(\))1607 4679 y(.)1607 4712 y(.)1607 4745 y(.)1516 4861 y Fu(')p Fw(\()p Fu(b)1659 4876 y Fq(t)1689 4861 y Fw(\))1731 4406 y Fm(1)1731 4582 y(C)1731 4642 y(C)1731 4705 y(A)1846 4667 y Fw(=)28 b Fu(D)2050 4406 y Fm(0)2050 4582 y(B)2050 4642 y(B)2050 4705 y(@)2137 4468 y Fu(b)2178 4483 y Fp(1)2137 4584 y Fu(b)2178 4599 y Fp(2)2164 4679 y Fw(.)2164 4712 y(.)2164 4745 y(.)2142 4861 y Fu(b)2183 4876 y Fq(t)2218 4406 y Fm(1)2218 4582 y(C)2218 4642 y(C)2218 4705 y(A)2322 4667 y Fu(;)94 5031 y Fw(where)34 b Fu(D)d Fw(=)c Fu(D)672 5046 y Fp(0)734 5031 y Fw(+)22 b Fu(D)913 5046 y Fp(1)985 5031 y Fw(is)32 b(an)g(in)m(v)m(ertible)g Fu(t)23 b Fr(\002)f Fu(t)33 b Fw(matrix)e(as)i(in)f(\()2463 5031 y SDict begin H.S end 2463 5031 a Fw(5.15)2636 4966 y SDict begin H.R end 2636 4966 a 2636 5031 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.15) cvn H.B /ANN pdfmark end 2636 5031 a Fw(\).)43 b(Therefore)34 b(w)m(e)g(ha)m(v)m(e)822 5200 y Fu(\013)28 b Fw(=)1016 5119 y Fm(\000)1062 5198 y Fu(a)1113 5213 y Fp(1)1235 5198 y Fu(a)1286 5213 y Fp(2)1409 5198 y Fr(\001)17 b(\001)g(\001)98 b Fu(a)1676 5213 y Fq(s)1713 5119 y Fm(\001)1758 5239 y Fq(M)1865 5200 y Fw(=)1969 5119 y Fm(\000)2014 5198 y Fu(')p Fw(\()p Fu(a)2167 5213 y Fp(1)2207 5198 y Fw(\))83 b Fu(')p Fw(\()p Fu(a)2481 5213 y Fp(2)2520 5198 y Fw(\))g Fr(\001)17 b(\001)g(\001)98 b Fu(')p Fw(\()p Fu(a)3010 5213 y Fq(s)3047 5198 y Fw(\))3085 5119 y Fm(\001)3131 5239 y Fq(M)3206 5220 y Ff(0)94 5200 y SDict begin H.S end 94 5200 a 94 5200 a SDict begin 17.5 H.A end 94 5200 a 94 5200 a SDict begin [ /View [/XYZ H.V] /Dest (equation.10.14) cvn H.B /DEST pdfmark end 94 5200 a Fw(\(10.14\))912 5361 y(=)28 b(\()1054 5280 y Fm(\000)1100 5359 y Fu(a)1151 5374 y Fp(1)1273 5359 y Fu(a)1324 5374 y Fp(2)1447 5359 y Fr(\001)17 b(\001)g(\001)98 b Fu(a)1714 5374 y Fq(s)1751 5280 y Fm(\001)1813 5361 y Fu(C)7 b Fw(\))1928 5376 y Fq(M)2003 5357 y Ff(0)2057 5361 y Fw(=)27 b Fu(\013)2223 5320 y Fn(0)2246 5361 y Fu(C)2316 5376 y Fq(M)2391 5357 y Ff(0)2418 5361 y Fu(;)94 5529 y Fw(and)33 b(similarly)-8 b(,)94 5696 y(\(10.15\))1635 5696 y SDict begin H.S end 1635 5696 a 1635 5696 a SDict begin 14 H.A end 1635 5696 a 1635 5696 a SDict begin [ /View [/XYZ H.V] /Dest (equation.10.15) cvn H.B /DEST pdfmark end 1635 5696 a Fu(\014)34 b Fw(=)27 b Fu(D)1908 5711 y Fq(M)1983 5692 y Ff(0)2010 5696 y Fu(\014)2071 5655 y Fn(0)2093 5696 y Fw(;)p eop end end %%Page: 45 45 TeXDict begin HPSdict begin 45 44 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.45) cvn H.B /DEST pdfmark end 94 218 a 1179 w Fs(QUIVERS)31 b(WITH)i(POTENTIALS)g(I)1102 b(45)94 417 y Fw(here)40 b Fu(C)382 432 y Fq(M)457 413 y Ff(0)522 417 y Fw(\(resp.)62 b Fu(D)903 432 y Fq(M)978 413 y Ff(0)1004 417 y Fw(\))39 b(is)f(understo)s(o)s(d)h(as)g(an)g Fu(R)q Fw(-mo)s(dule)d(automorphism)h(of)h Fu(M)3299 381 y Fn(0)3289 442 y Fp(in)3390 417 y Fw(=)g Fu(M)3598 432 y Fp(in)94 533 y Fw(\(resp.)45 b(of)32 b Fu(M)592 497 y Fn(0)582 558 y Fp(out)716 533 y Fw(=)c Fu(M)914 548 y Fp(out)1020 533 y Fw(\).)194 649 y(T)-8 b(urning)32 b(to)h(the)g(maps)f Fu(\015)37 b Fw(and)c Fu(\015)1442 613 y Fn(0)1465 649 y Fw(,)g(w)m(e)h(claim)c(that)i(they)i(are)e(related)g(b)m(y)94 809 y(\(10.16\))1554 809 y SDict begin H.S end 1554 809 a 1554 809 a SDict begin 14 H.A end 1554 809 a 1554 809 a SDict begin [ /View [/XYZ H.V] /Dest (equation.10.16) cvn H.B /DEST pdfmark end 1554 809 a Fu(\015)1610 768 y Fn(0)1661 809 y Fw(=)c Fu(C)1835 824 y Fq(M)1910 805 y Ff(0)1936 809 y Fu(\015)5 b(D)2073 824 y Fq(M)2148 805 y Ff(0)2174 809 y Fu(:)94 969 y Fw(T)-8 b(o)33 b(see)h(this,)e(w)m(e)i(use)f(\()963 969 y SDict begin H.S end 963 969 a Fw(10.4)1137 905 y SDict begin H.R end 1137 905 a 1137 969 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.4) cvn H.B /ANN pdfmark end 1137 969 a Fw(\))f(and)h(\()1435 969 y SDict begin H.S end 1435 969 a Fw(3.6)1559 905 y SDict begin H.R end 1559 905 a 1559 969 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.3.6) cvn H.B /ANN pdfmark end 1559 969 a Fw(\))g(to)f(write)1113 1129 y Fu(\015)1169 1088 y Fn(0)1164 1154 y Fq(p;q)1285 1129 y Fw(=)27 b(\()p Fu(@)1477 1145 y Fp([)p Fq(b)1527 1153 y Fg(q)1562 1145 y Fq(a)1599 1153 y Fg(p)1635 1145 y Fp(])1659 1129 y Fu(')p Fw(\()p Fu(S)6 b Fw(\)\))1903 1144 y Fq(M)1978 1125 y Ff(0)94 1129 y SDict begin H.S end 94 1129 a 94 1129 a SDict begin 17.5 H.A end 94 1129 a 94 1129 a SDict begin [ /View [/XYZ H.V] /Dest (equation.10.17) cvn H.B /DEST pdfmark end 94 1129 a Fw(\(10.17\))1285 1310 y(=)27 b(\()1426 1216 y Fm(X)1483 1425 y Fq(c)1587 1310 y Fw(\001)1668 1326 y Fp([)p Fq(b)1718 1334 y Fg(q)1752 1326 y Fq(a)1789 1334 y Fg(p)1826 1326 y Fp(])1849 1310 y Fw(\()p Fu(')p Fw(\()p Fu(c)p Fw(\)\))p Fh(\003)p Fu(')p Fw(\()p Fu(@)2337 1325 y Fq(c)2373 1310 y Fu(S)6 b Fw(\)\))2515 1325 y Fq(M)2590 1306 y Ff(0)2616 1310 y Fu(;)94 1584 y Fw(where)41 b(the)e(sum)g(is)f(o)m(v)m(er)i(all)d(arro)m(ws)i Fu(c)g Fw(in)1773 1559 y Fm(e)1750 1584 y Fu(A)1825 1596 y Fp(^)1823 1614 y Fq(k)r(;)1884 1596 y Fp(^)1882 1614 y Fq(k)1924 1584 y Fw(.)63 b(If)38 b Fu(c)h Fw(is)g(one)g(of)f(the)h (arro)m(ws)h(in)e Fu(A)h Fw(then)g(b)m(y)94 1709 y(\()132 1709 y SDict begin H.S end 132 1709 a Fw(10.11)355 1645 y SDict begin H.R end 355 1645 a 355 1709 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.11) cvn H.B /ANN pdfmark end 355 1709 a Fw(\))32 b(w)m(e)i(ha)m(v)m(e)1359 1829 y Fu(')p Fw(\()p Fu(@)1512 1844 y Fq(c)1547 1829 y Fu(S)6 b Fw(\))1651 1844 y Fq(M)1726 1825 y Ff(0)1780 1829 y Fw(=)27 b(\()p Fu(@)1972 1844 y Fq(c)2007 1829 y Fu(S)6 b Fw(\))2111 1844 y Fq(M)2218 1829 y Fw(=)27 b(0;)94 1968 y(remem)m(b)s(ering)g(the)h(de\014nition)f(\()1305 1968 y SDict begin H.S end 1305 1968 a Fw(3.3)1429 1904 y SDict begin H.R end 1429 1904 a 1429 1968 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.3.3) cvn H.B /ANN pdfmark end 1429 1968 a Fw(\),)i(w)m(e)g(see)g(that)e Fu(c)h Fw(do)s(es)g(not)g(con)m(tribute)f(to)h(\()3091 1968 y SDict begin H.S end 3091 1968 a Fw(10.17)3313 1904 y SDict begin H.R end 3313 1904 a 3313 1968 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.17) cvn H.B /ANN pdfmark end 3313 1968 a Fw(\).)42 b(Th)m(us,)94 2085 y(w)m(e)34 b(ha)m(v)m(e)94 2245 y(\(10.18\))931 2245 y SDict begin H.S end 931 2245 a 931 2245 a SDict begin 14 H.A end 931 2245 a 931 2245 a SDict begin [ /View [/XYZ H.V] /Dest (equation.10.18) cvn H.B /DEST pdfmark end 931 2245 a Fu(\015)987 2204 y Fn(0)982 2269 y Fq(p;q)1102 2245 y Fw(=)28 b(\()1244 2150 y Fm(X)1249 2363 y Fq(p)1285 2344 y Ff(0)1307 2363 y Fq(;q)1361 2344 y Ff(0)1404 2245 y Fw(\001)1485 2260 y Fp([)p Fq(b)1535 2268 y Fg(q)1570 2260 y Fq(a)1607 2268 y Fg(p)1643 2260 y Fp(])1667 2245 y Fw(\()p Fu(')p Fw(\()p Fu(b)1848 2260 y Fq(q)1882 2241 y Ff(0)1909 2245 y Fu(a)1960 2260 y Fq(p)1996 2241 y Ff(0)2022 2245 y Fw(\)\))p Fh(\003)p Fu(')p Fw(\()p Fu(@)2328 2260 y Fp([)p Fq(b)2378 2279 y Fg(q)2408 2265 y Ff(0)2435 2260 y Fq(a)2472 2279 y Fg(p)2504 2265 y Ff(0)2531 2260 y Fp(])2555 2245 y Fu(S)6 b Fw(\)\))2697 2260 y Fq(M)2772 2241 y Ff(0)2798 2245 y Fu(:)94 2512 y Fw(Remem)m(b)s(ering)36 b(\()756 2512 y SDict begin H.S end 756 2512 a Fw(3.2)880 2448 y SDict begin H.R end 880 2448 a 880 2512 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.3.2) cvn H.B /ANN pdfmark end 880 2512 a Fw(\),)i(and)e(using)h(\()1474 2512 y SDict begin H.S end 1474 2512 a Fw(10.12)1696 2448 y SDict begin H.R end 1696 2448 a 1696 2512 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.12) cvn H.B /ANN pdfmark end 1696 2512 a Fw(\))f(and)h(\()2002 2512 y SDict begin H.S end 2002 2512 a Fw(10.13)2224 2448 y SDict begin H.R end 2224 2448 a 2224 2512 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.13) cvn H.B /ANN pdfmark end 2224 2512 a Fw(\),)g(w)m(e)h(see)g(that) e(the)h(summand)f(with)94 2629 y(\()p Fu(p)181 2592 y Fn(0)205 2629 y Fu(;)17 b(q)296 2592 y Fn(0)318 2629 y Fw(\))28 b(=)g(\()p Fu(p;)17 b(q)t Fw(\))32 b(in)f(\()887 2629 y SDict begin H.S end 887 2629 a Fw(10.18)1109 2564 y SDict begin H.R end 1109 2564 a 1109 2629 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.18) cvn H.B /ANN pdfmark end 1109 2629 a Fw(\))i(con)m(tains)g(among)e(its)h(terms)g(the)h (\()p Fu(p;)17 b(q)t Fw(\)-en)m(try)33 b(of)f(the)h(matrix)1206 2789 y(\()p Fu(C)7 b(')p Fw(\()p Fu(@)1474 2804 y Fp([)p Fq(b)1524 2812 y Fg(q)1558 2804 y Fq(a)1595 2812 y Fg(p)1632 2804 y Fp(])1656 2789 y Fu(S)f Fw(\))p Fu(D)s Fw(\))1882 2804 y Fq(M)1957 2785 y Ff(0)2010 2789 y Fw(=)27 b Fu(C)2183 2804 y Fq(M)2258 2785 y Ff(0)2285 2789 y Fu(\015)5 b(D)2422 2804 y Fq(M)2497 2785 y Ff(0)2523 2789 y Fu(:)94 2949 y Fw(Th)m(us,)35 b(to)e(pro)m(v)m(e)i(\()791 2949 y SDict begin H.S end 791 2949 a Fw(10.16)1013 2885 y SDict begin H.R end 1013 2885 a 1013 2949 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.16) cvn H.B /ANN pdfmark end 1013 2949 a Fw(\),)e(it)f(remains)g(to)h(sho)m(w)h(that)f(the)h(rest)f(of)g (the)h(terms)f(in)f(\()3211 2949 y SDict begin H.S end 3211 2949 a Fw(10.18)3433 2885 y SDict begin H.R end 3433 2885 a 3433 2949 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.18) cvn H.B /ANN pdfmark end 3433 2949 a Fw(\))h(add)94 3065 y(up)k(to)f(0.)54 b(Again)35 b(using)h(the)h(de\014nitions)f(\() 1725 3065 y SDict begin H.S end 1725 3065 a Fw(3.2)1849 3001 y SDict begin H.R end 1849 3001 a 1849 3065 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.3.2) cvn H.B /ANN pdfmark end 1849 3065 a Fw(\))g(and)h(\()2155 3065 y SDict begin H.S end 2155 3065 a Fw(3.3)2279 3001 y SDict begin H.R end 2279 3001 a 2279 3065 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.3.3) cvn H.B /ANN pdfmark end 2279 3065 a Fw(\),)h(w)m(e)f(can)f(rewrite)h(the)f(rest)h(of)f(the)94 3181 y(sum)d(in)f(\()453 3181 y SDict begin H.S end 453 3181 a Fw(10.18)675 3117 y SDict begin H.R end 675 3117 a 675 3181 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.18) cvn H.B /ANN pdfmark end 675 3181 a Fw(\))g(as)h Fu(S)925 3196 y Fp(1)987 3181 y Fw(+)22 b Fu(S)1145 3196 y Fp(2)1184 3181 y Fw(,)33 b(where)917 3363 y Fu(S)977 3378 y Fp(1)1044 3363 y Fw(=)28 b(\()1186 3268 y Fm(X)1191 3481 y Fq(p)1227 3462 y Ff(0)1249 3481 y Fq(;q)1303 3462 y Ff(0)1346 3363 y Fw(\001)1427 3378 y Fp([)p Fq(b)1477 3386 y Fg(q)1512 3378 y Fq(a)1549 3386 y Fg(p)1585 3378 y Fp(])1609 3363 y Fw(\()p Fu(')p Fw(\()p Fu(b)1790 3378 y Fq(q)1824 3359 y Ff(0)1851 3363 y Fw(\)\))p Fh(\003)p Fu(')p Fw(\()p Fu(a)2157 3378 y Fq(p)2193 3359 y Ff(0)2241 3363 y Fr(\001)22 b Fu(@)2342 3378 y Fp([)p Fq(b)2392 3397 y Fg(q)2422 3383 y Ff(0)2449 3378 y Fq(a)2486 3397 y Fg(p)2518 3383 y Ff(0)2545 3378 y Fp(])2569 3363 y Fu(S)6 b Fw(\)\))2711 3378 y Fq(M)2786 3359 y Ff(0)2812 3363 y Fu(;)917 3663 y(S)977 3678 y Fp(2)1044 3663 y Fw(=)28 b(\()1186 3569 y Fm(X)1191 3781 y Fq(p)1227 3763 y Ff(0)1249 3781 y Fq(;q)1303 3763 y Ff(0)1346 3663 y Fw(\001)1427 3679 y Fp([)p Fq(b)1477 3687 y Fg(q)1512 3679 y Fq(a)1549 3687 y Fg(p)1585 3679 y Fp(])1609 3663 y Fw(\()p Fu(')p Fw(\()p Fu(a)1800 3678 y Fq(p)1836 3660 y Ff(0)1862 3663 y Fw(\)\))p Fh(\003)p Fu(')p Fw(\()p Fu(@)2168 3679 y Fp([)p Fq(b)2218 3698 y Fg(q)2248 3684 y Ff(0)2275 3679 y Fq(a)2312 3698 y Fg(p)2344 3684 y Ff(0)2371 3679 y Fp(])2395 3663 y Fu(S)g Fr(\001)22 b Fu(b)2574 3678 y Fq(q)2608 3660 y Ff(0)2635 3663 y Fw(\)\))2711 3678 y Fq(M)2786 3660 y Ff(0)2812 3663 y Fu(:)94 3905 y Fw(It)33 b(remains)f(to)g(observ)m(e)i(that)1001 4065 y Fu(S)1061 4080 y Fp(1)1128 4065 y Fw(=)27 b(\()1269 3971 y Fm(X)1313 4183 y Fq(q)1347 4164 y Ff(0)1430 4065 y Fw(\001)1511 4081 y Fp([)p Fq(b)1561 4089 y Fg(q)1595 4081 y Fq(a)1632 4089 y Fg(p)1669 4081 y Fp(])1693 4065 y Fw(\()p Fu(')p Fw(\()p Fu(b)1874 4080 y Fq(q)1908 4061 y Ff(0)1934 4065 y Fw(\)\))p Fh(\003)p Fu(')p Fw(\()p Fu(@)2240 4080 y Fq(b)2270 4099 y Fg(q)2300 4085 y Ff(0)2332 4065 y Fu(S)6 b Fw(\)\))2474 4080 y Fq(M)2549 4061 y Ff(0)2603 4065 y Fw(=)27 b(0)94 4333 y(since)33 b Fu(')p Fw(\()p Fu(@)486 4348 y Fq(b)516 4367 y Fg(q)546 4353 y Ff(0)578 4333 y Fu(S)6 b Fw(\))682 4348 y Fq(M)757 4329 y Ff(0)811 4333 y Fw(=)27 b(\()p Fu(@)1003 4348 y Fq(b)1033 4367 y Fg(q)1063 4353 y Ff(0)1095 4333 y Fu(S)6 b Fw(\))1199 4348 y Fq(M)1305 4333 y Fw(=)28 b(0;)k(and)h (similarly)-8 b(,)977 4533 y Fu(S)1037 4548 y Fp(2)1104 4533 y Fw(=)28 b(\()1246 4439 y Fm(X)1289 4651 y Fq(p)1325 4632 y Ff(0)1406 4533 y Fw(\001)1487 4549 y Fp([)p Fq(b)1537 4557 y Fg(q)1572 4549 y Fq(a)1609 4557 y Fg(p)1645 4549 y Fp(])1669 4533 y Fw(\()p Fu(')p Fw(\()p Fu(a)1860 4548 y Fq(p)1896 4529 y Ff(0)1922 4533 y Fw(\)\))p Fh(\003)p Fu(')p Fw(\()p Fu(@)2228 4548 y Fq(a)2265 4567 y Fg(p)2297 4553 y Ff(0)2329 4533 y Fu(S)6 b Fw(\)\))2471 4548 y Fq(M)2546 4529 y Ff(0)2600 4533 y Fw(=)27 b(0)p Fu(:)194 4801 y Fw(In)33 b(view)g(of)f(\()690 4801 y SDict begin H.S end 690 4801 a Fw(10.14)912 4737 y SDict begin H.R end 912 4737 a 912 4801 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.14) cvn H.B /ANN pdfmark end 912 4801 a Fw(\),)h(\()1048 4801 y SDict begin H.S end 1048 4801 a Fw(10.15)1270 4737 y SDict begin H.R end 1270 4737 a 1270 4801 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.15) cvn H.B /ANN pdfmark end 1270 4801 a Fw(\))f(and)h(\()1568 4801 y SDict begin H.S end 1568 4801 a Fw(10.16)1790 4737 y SDict begin H.R end 1790 4737 a 1790 4801 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.16) cvn H.B /ANN pdfmark end 1790 4801 a Fw(\),)g(w)m(e)g(ha)m(v)m(e)903 4962 y SDict begin H.S end 903 4962 a 903 4962 a SDict begin 14 H.A end 903 4962 a 903 4962 a SDict begin [ /View [/XYZ H.V] /Dest (equation.10.19) cvn H.B /DEST pdfmark end 903 4962 a 17 w Fw(k)m(er)17 b Fu(\013)28 b Fw(=)83 b Fu(C)1392 4921 y Fn(\000)p Fp(1)1385 4990 y Fq(M)1460 4971 y Ff(0)1487 4962 y Fw(\(k)m(er)17 b Fu(\013)1734 4926 y Fn(0)1757 4962 y Fw(\))p Fu(;)114 b Fw(im)31 b Fu(\013)84 b Fw(=)27 b(im)k Fu(\013)2530 4921 y Fn(0)2553 4962 y Fu(;)921 5107 y Fw(k)m(er)18 b Fu(\014)33 b Fw(=)86 b(k)m(er)18 b Fu(\014)1526 5071 y Fn(0)1549 5107 y Fu(;)359 b Fw(im)31 b Fu(\014)92 b Fw(=)27 b Fu(D)2407 5122 y Fq(M)2482 5103 y Ff(0)2509 5107 y Fw(\(im)k Fu(\014)2749 5066 y Fn(0)2772 5107 y Fw(\))p Fu(;)-2743 b Fw(\(10.19\))926 5254 y(k)m(er)18 b Fu(\015)32 b Fw(=)84 b Fu(D)1397 5269 y Fq(M)1472 5250 y Ff(0)1499 5254 y Fw(\(k)m(er)17 b Fu(\015)1739 5217 y Fn(0)1763 5254 y Fw(\))p Fu(;)114 b Fw(im)31 b Fu(\015)89 b Fw(=)27 b Fu(C)2403 5212 y Fn(\000)p Fp(1)2396 5282 y Fq(M)2471 5263 y Ff(0)2498 5254 y Fw(\(im)k Fu(\015)2733 5212 y Fn(0)2756 5254 y Fw(\))p Fu(:)94 5422 y Fw(Recall)36 b(that)g(the)i(spaces)p 1084 5342 105 4 v 38 w Fu(M)48 b Fw(and)p 1420 5342 79 4 v 37 w Fu(V)58 b Fw(in)36 b(the)h(decorated)h (represen)m(tation)k Fm(e)-59 b Fu(\026)2975 5437 y Fq(k)3017 5422 y Fw(\()p Fr(M)p Fw(\))35 b(=)g(\()p 3397 5342 105 4 v Fu(M)10 b(;)p 3545 5342 79 4 v 17 w(V)21 b Fw(\))94 5552 y(of)39 b(\()273 5527 y Fm(e)250 5552 y Fu(A;)381 5527 y Fm(e)367 5552 y Fu(S)6 b Fw(\))39 b(are)h(giv)m(en)f(b)m(y)i(\() 1122 5552 y SDict begin H.S end 1122 5552 a Fw(10.6)1295 5488 y SDict begin H.R end 1295 5488 a 1295 5552 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.6) cvn H.B /ANN pdfmark end 1295 5552 a Fw(\))e(and)h(\()1607 5552 y SDict begin H.S end 1607 5552 a Fw(10.7)1780 5488 y SDict begin H.R end 1780 5488 a 1780 5552 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.7) cvn H.B /ANN pdfmark end 1780 5552 a Fw(\).)64 b(W)-8 b(e)40 b(express)i(the)d(decorated)i (represen)m(tation)99 5696 y Fm(e)-60 b Fu(\026)153 5711 y Fq(k)196 5696 y Fw(\()p Fr(M)354 5660 y Fn(0)377 5696 y Fw(\))27 b(=)h(\()p 584 5616 129 4 v Fu(M)688 5667 y Fn(0)712 5696 y Fu(;)p 756 5616 102 4 v 17 w(V)834 5667 y Fn(0)857 5696 y Fw(\))j(of)f(\()1095 5671 y Fm(e)1073 5696 y Fu(A;)1198 5668 y Fd(])1190 5696 y Fu(')p Fw(\()p Fu(S)6 b Fw(\))o(\))30 b(in)g(the)h(same)f(w)m(a)m(y)-8 b(,)33 b(with)d(the)h(maps)f Fu(\013)q(;)17 b(\014)35 b Fw(and)c Fu(\015)k Fw(replaced)p eop end end %%Page: 46 46 TeXDict begin HPSdict begin 46 45 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.46) cvn H.B /DEST pdfmark end 94 218 a Fs(46)492 b(HARM)25 b(DERKSEN,)f(JERZY)i(WEYMAN,)f(AND)f(ANDREI)g(ZELEVINSKY)94 422 y Fw(b)m(y)42 b Fu(\013)301 386 y Fn(0)324 422 y Fu(;)17 b(\014)429 386 y Fn(0)491 422 y Fw(and)41 b Fu(\015)745 386 y Fn(0)768 422 y Fw(.)67 b(In)40 b(particular,)h(w)m(e)g(ha)m(v)m (e)p 1862 342 102 4 v 42 w Fu(V)1940 393 y Fn(0)2004 422 y Fw(=)p 2121 342 79 4 v 41 w Fu(V)21 b Fw(,)43 b(and)p 2466 342 129 4 v 40 w Fu(M)2570 393 y Fn(0)2594 437 y Fq(i)2663 422 y Fw(=)p 2780 342 105 4 v 41 w Fu(M)2884 437 y Fq(i)2953 422 y Fw(=)e Fu(M)3164 437 y Fq(i)3233 422 y Fw(for)f Fu(i)h Fr(6)p Fw(=)f Fu(k)s Fw(.)94 538 y(T)-8 b(o)32 b(sp)s(ecify)g(the)g(actions)f(of)g Fu(R)q Fr(hh)p Fu(A)p Fr(ii)g Fw(in)p 1613 458 V 30 w Fu(M)43 b Fw(and)p 1938 458 129 4 v 31 w Fu(M)2042 509 y Fn(0)2066 538 y Fw(,)32 b(w)m(e)g(need)h(to)e(c)m(ho)s(ose)i(the)e(splitting)f (data)94 654 y(\()p Fu(\032;)17 b(\033)t Fw(\))26 b(and)g(\()p Fu(\032)620 618 y Fn(0)644 654 y Fu(;)17 b(\033)747 618 y Fn(0)770 654 y Fw(\))26 b(as)g(in)f(\()1092 654 y SDict begin H.S end 1092 654 a Fw(10.8)1266 590 y SDict begin H.R end 1266 590 a 1266 654 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.8) cvn H.B /ANN pdfmark end 1266 654 a Fw(\))h(and)g(\()1551 654 y SDict begin H.S end 1551 654 a Fw(10.9)1724 590 y SDict begin H.R end 1724 590 a 1724 654 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.9) cvn H.B /ANN pdfmark end 1724 654 a Fw(\).)41 b(Note)26 b(that,)i(in)d(view)h(of)g(\()2761 654 y SDict begin H.S end 2761 654 a Fw(10.19)2983 590 y SDict begin H.R end 2983 590 a 2983 654 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.19) cvn H.B /ANN pdfmark end 2983 654 a Fw(\),)h(w)m(e)g(can)f(c)m(ho)s (ose)94 818 y(\(10.20\))1176 818 y SDict begin H.S end 1176 818 a 1176 818 a SDict begin 14 H.A end 1176 818 a 1176 818 a SDict begin [ /View [/XYZ H.V] /Dest (equation.10.20) cvn H.B /DEST pdfmark end 1176 818 a Fu(\032)1226 776 y Fn(0)1278 818 y Fw(=)h Fu(D)1465 776 y Fn(\000)p Fp(1)1462 846 y Fq(M)1537 827 y Ff(0)1564 818 y Fu(\032D)1695 833 y Fq(M)1770 814 y Ff(0)1796 818 y Fu(;)114 b(\033)1996 776 y Fn(0)2047 818 y Fw(=)28 b Fu(C)2221 833 y Fq(M)2296 814 y Ff(0)2322 818 y Fu(\033)t(C)2458 776 y Fn(\000)p Fp(1)2451 846 y Fq(M)2526 827 y Ff(0)2552 818 y Fw(;)94 984 y(here)i(with)e(some)g(abuse)i(of)e(notation)f(w)m(e)i(use)h(the)f (same)f(notation)f Fu(C)2692 943 y Fn(\000)p Fp(1)2685 1012 y Fq(M)2760 993 y Ff(0)2815 984 y Fw(for)h(the)h(isomorphism)94 1100 y(k)m(er)18 b Fu(\013)304 1064 y Fn(0)355 1100 y Fr(!)27 b Fw(k)m(er)18 b Fu(\013)33 b Fw(and)g(the)g(induced)g (isomorphism)d(k)m(er)18 b Fu(\013)2225 1064 y Fn(0)2248 1100 y Fu(=)p Fw(im)30 b Fu(\015)2493 1064 y Fn(0)2544 1100 y Fr(!)d Fw(k)m(er)18 b Fu(\013)q(=)p Fw(im)31 b Fu(\015)5 b Fw(.)194 1216 y(Ev)m(erything)36 b(is)e(no)m(w)h(in)f (place)g(for)g(de\014ning)h(the)g(desired)g(righ)m(t-equiv)-5 b(alence)33 b(\()12 b Fm(b)-67 b Fu(';)17 b( )t(;)g(\021)t Fw(\))34 b(b)s(e-)94 1347 y(t)m(w)m(een)j Fm(e)-59 b Fu(\026)428 1362 y Fq(k)470 1347 y Fw(\()p Fr(M)p Fw(\))30 b(and)k Fm(e)-59 b Fu(\026)942 1362 y Fq(k)984 1347 y Fw(\()p Fr(M)1142 1311 y Fn(0)1165 1347 y Fw(\))30 b(\(see)i (De\014nition)1878 1347 y SDict begin H.S end 1878 1347 a Fw(10.2)2051 1282 y SDict begin H.R end 2051 1282 a 2051 1347 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.10.2) cvn H.B /ANN pdfmark end 2051 1347 a Fw(\).)43 b(First)29 b(of)h(all,)f(w)m(e)i(de\014ne)44 b Fm(b)-67 b Fu(')27 b Fw(:)h Fu(R)q Fr(hh)3406 1322 y Fm(e)3384 1347 y Fu(A)p Fr(ii)f(!)94 1491 y Fu(R)q Fr(hh)269 1466 y Fm(e)247 1491 y Fu(A)p Fr(ii)d Fw(as)h(the)g(righ)m(t-equiv)-5 b(alence)24 b(b)s(et)m(w)m(een)i(\()1866 1466 y Fm(e)1843 1491 y Fu(A;)1974 1466 y Fm(e)1960 1491 y Fu(S)6 b Fw(\))24 b(and)h(\()2331 1466 y Fm(e)2308 1491 y Fu(A;)2434 1462 y Fd(])2425 1491 y Fu(')p Fw(\()p Fu(S)6 b Fw(\))24 b(constructed)i(in) e(the)h(pro)s(of)94 1607 y(of)33 b(Lemma)554 1607 y SDict begin H.S end 554 1607 a Fw(5.3)678 1543 y SDict begin H.R end 678 1543 a 678 1607 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.5.3) cvn H.B /ANN pdfmark end 678 1607 a Fw(.)44 b(In)33 b(particular,)e(w)m(e)i(ha)m(v)m(e)203 1700 y Fm(0)203 1875 y(B)203 1935 y(B)203 1999 y(@)302 1762 y(b)-67 b Fu(')p Fw(\()p Fu(a)443 1726 y Fq(?)443 1786 y Fp(1)482 1762 y Fw(\))302 1878 y Fm(b)g Fu(')p Fw(\()p Fu(a)443 1842 y Fq(?)443 1903 y Fp(2)482 1878 y Fw(\))391 1972 y(.)391 2006 y(.)391 2039 y(.)302 2155 y Fm(b)g Fu(')p Fw(\()p Fu(a)443 2119 y Fq(?)443 2180 y(s)482 2155 y Fw(\))520 1700 y Fm(1)520 1875 y(C)520 1935 y(C)520 1999 y(A)635 1960 y Fw(=)27 b Fu(C)815 1919 y Fn(\000)p Fp(1)926 1700 y Fm(0)926 1875 y(B)926 1935 y(B)926 1999 y(@)1014 1762 y Fu(a)1065 1726 y Fq(?)1065 1786 y Fp(1)1014 1878 y Fu(a)1065 1842 y Fq(?)1065 1903 y Fp(2)1045 1972 y Fw(.)1045 2006 y(.)1045 2039 y(.)1014 2155 y Fu(a)1065 2119 y Fq(?)1065 2180 y(s)1104 1700 y Fm(1)1104 1875 y(C)1104 1935 y(C)1104 1999 y(A)1208 1960 y Fu(;)1349 1879 y Fm(\000)1407 1958 y(b)-67 b Fu(')p Fw(\()p Fu(b)1538 1922 y Fq(?)1538 1983 y Fp(1)1578 1958 y Fw(\))95 b Fm(b)-67 b Fu(')o Fw(\()p Fu(b)1841 1922 y Fq(?)1841 1983 y Fp(2)1881 1958 y Fw(\))83 b Fr(\001)17 b(\001)g(\001)111 b Fm(b)-68 b Fu(')p Fw(\()p Fu(b)2361 1922 y Fq(?)2361 1983 y(t)2401 1958 y Fw(\))2439 1879 y Fm(\001)2512 1960 y Fw(=)2616 1879 y Fm(\000)2661 1958 y Fu(b)2702 1922 y Fq(?)2702 1983 y Fp(1)2825 1958 y Fu(b)2866 1922 y Fq(?)2866 1983 y Fp(2)2989 1958 y Fr(\001)17 b(\001)g(\001)98 b Fu(b)3246 1922 y Fq(?)3246 1983 y(t)3286 1879 y Fm(\001)3348 1960 y Fu(D)3432 1919 y Fn(\000)p Fp(1)3526 1960 y Fu(:)194 2332 y Fw(Next)26 b(w)m(e)h(de\014ne)f Fu( )32 b Fw(:)p 986 2252 105 4 v 28 w Fu(M)38 b Fr(!)p 1245 2252 129 4 v 27 w Fu(M)1349 2303 y Fn(0)1399 2332 y Fw(as)25 b(the)h(iden)m(tit)m(y)f(map)g(on)g Fr(\010)2440 2347 y Fq(i)p Fn(6)p Fp(=)p Fq(k)p 2562 2252 105 4 v 2562 2332 a Fu(M)2667 2347 y Fq(i)2723 2332 y Fw(=)i Fr(\010)2903 2347 y Fq(i)p Fn(6)p Fp(=)p Fq(k)3025 2332 y Fu(M)3119 2347 y Fq(i)3175 2332 y Fw(=)h Fr(\010)3356 2347 y Fq(i)p Fn(6)p Fp(=)p Fq(k)p 3478 2252 129 4 v 3478 2332 a Fu(M)3582 2303 y Fn(0)3606 2347 y Fq(i)3634 2332 y Fw(,)94 2456 y(and)33 b(the)g(restriction)f Fu( )t Fr(j)p 1014 2428 75 3 v 26 x Fq(M)1088 2494 y Fg(k)1158 2456 y Fw(:)p 1213 2376 105 4 v 28 w Fu(M)1318 2471 y Fq(k)1388 2456 y Fr(!)p 1515 2376 129 4 v 27 w Fu(M)1619 2427 y Fn(0)1643 2471 y Fq(k)1719 2456 y Fw(giv)m(en)g(b)m(y)i(the)f (blo)s(c)m(k-diagonal)c(matrix)94 2812 y(\(10.21\))1207 2812 y SDict begin H.S end 1207 2812 a 1207 2812 a SDict begin 14 H.A end 1207 2812 a 1207 2812 a SDict begin [ /View [/XYZ H.V] /Dest (equation.10.21) cvn H.B /DEST pdfmark end 1207 2812 a Fu( )t Fr(j)p 1302 2784 75 3 v 26 x Fq(M)1376 2850 y Fg(k)1446 2812 y Fw(=)1550 2552 y Fm(0)1550 2727 y(B)1550 2787 y(B)1550 2851 y(@)1637 2637 y Fu(D)1721 2596 y Fn(\000)p Fp(1)1718 2666 y Fq(M)1793 2647 y Ff(0)1964 2637 y Fw(0)205 b(0)145 b(0)1704 2754 y(0)150 b Fu(C)1973 2769 y Fq(M)2048 2750 y Ff(0)2218 2754 y Fw(0)145 b(0)1704 2870 y(0)211 b(0)144 b Fu(C)2227 2885 y Fq(M)2302 2866 y Ff(0)2412 2870 y Fw(0)1704 2986 y(0)211 b(0)205 b(0)144 b Fu(I)2462 2552 y Fm(1)2462 2727 y(C)2462 2787 y(C)2462 2851 y(A)94 3168 y Fw(\(this)33 b(is)f(w)m(ell-de\014ned)h(in)f(view)h (of)g(\()1444 3168 y SDict begin H.S end 1444 3168 a Fw(10.19)1666 3104 y SDict begin H.R end 1666 3104 a 1666 3168 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.19) cvn H.B /ANN pdfmark end 1666 3168 a Fw(\)\).)44 b(Finally)-8 b(,)30 b(w)m(e)k(de\014ne)g Fu(\021)e Fw(:)p 2725 3088 79 4 v 28 w Fu(V)49 b Fr(!)p 2959 3088 102 4 v 28 w Fu(V)3038 3140 y Fn(0)3094 3168 y Fw(simply)31 b(as)i(the)94 3285 y(iden)m(tit)m(y)g(map.)194 3409 y(The)h(only)f(thing)g(to)g(c)m(hec)m (k)j(is)c(the)i(equalit)m(y)f Fu( )27 b Fr(\016)c Fu(c)p 2094 3381 79 3 v 26 x Fq(M)2202 3409 y Fw(=)29 b Fu(')p Fw(\()p Fu(c)p Fw(\))p 2489 3381 102 3 v 28 x Fq(M)2564 3418 y Ff(0)2612 3409 y Fr(\016)23 b Fu( )37 b Fw(for)c(an)m(y)h(arro)m (w)g Fu(c)29 b Fr(2)3584 3384 y Fm(e)3561 3409 y Fu(A)p Fw(.)94 3526 y(And)f(the)f(only)g(case)g(that)g(ma)m(y)g(require)g (some)g(consideration)f(is)g(when)i Fu(c)f Fw(is)g(one)g(of)f(the)i (arro)m(ws)94 3642 y Fu(a)145 3606 y Fq(?)145 3667 y(p)218 3642 y Fw(or)k Fu(b)378 3606 y Fq(?)378 3667 y(q)418 3642 y Fw(.)43 b(Unra)m(v)m(eling)33 b(the)g(de\014nitions,)f(it)f (su\016ces)k(to)d(sho)m(w)i(that)p 1033 3742 61 4 v 1033 3823 a Fu(\014)g Fw(=)27 b Fu(C)1302 3781 y Fn(\000)p Fp(1)1295 3851 y Fq(M)1370 3832 y Ff(0)p 1396 3742 84 4 v 1396 3823 a Fu(\014)1457 3794 y Fn(0)1502 3823 y Fr(\016)22 b Fu( )t Fr(j)p 1669 3794 75 3 v 26 x Fq(M)1744 3860 y Fg(k)1786 3823 y Fu(;)114 b( )t Fr(j)p 2022 3794 V 26 x Fq(M)2097 3860 y Fg(k)2161 3823 y Fr(\016)p 2233 3768 63 4 v 22 w Fu(\013)29 b Fw(=)p 2427 3745 86 4 v 27 w Fu(\013)2490 3794 y Fn(0)2513 3823 y Fu(D)2597 3781 y Fn(\000)p Fp(1)2594 3851 y Fq(M)2669 3832 y Ff(0)2695 3823 y Fu(:)94 3989 y Fw(But)i(this)g(is)f(an)g(immediate)e (consequence)34 b(of)c(the)h(de\014nitions)g(\()2512 3989 y SDict begin H.S end 2512 3989 a Fw(10.21)2734 3925 y SDict begin H.R end 2734 3925 a 2734 3989 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.21) cvn H.B /ANN pdfmark end 2734 3989 a Fw(\))f(and)h(\()3028 3989 y SDict begin H.S end 3028 3989 a Fw(10.10)3250 3925 y SDict begin H.R end 3250 3925 a 3250 3989 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.10) cvn H.B /ANN pdfmark end 3250 3989 a Fw(\))g(\(w)m(e)g(also)94 4112 y(need)f(an)e(analogue)f(of)h(\() 996 4112 y SDict begin H.S end 996 4112 a Fw(10.10)1218 4047 y SDict begin H.R end 1218 4047 a 1218 4112 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.10) cvn H.B /ANN pdfmark end 1218 4112 a Fw(\))h(for)f(the)g(maps)p 1844 4030 84 4 v 28 w Fu(\014)1905 4083 y Fn(0)1957 4112 y Fw(and)p 2142 4034 86 4 v 28 w Fu(\013)2205 4083 y Fn(0)2228 4112 y Fw(,)h(using)f(the)h(splitting)d(data)i(\()3336 4112 y SDict begin H.S end 3336 4112 a Fw(10.20)3558 4047 y SDict begin H.R end 3558 4047 a 3558 4112 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.20) cvn H.B /ANN pdfmark end 3558 4112 a Fw(\)\).)94 4228 y(This)33 b(completes)g(the)g(pro)s (of)e(of)h(Prop)s(osition)1826 4228 y SDict begin H.S end 1826 4228 a Fw(10.10)2048 4164 y SDict begin H.R end 2048 4164 a 2048 4228 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.10.10) cvn H.B /ANN pdfmark end 2048 4228 a Fw(.)1509 b Fh(\003)194 4400 y Fw(Note)51 b(that)g(in)g(the)g(ab)s(o)m(v)m(e)h (treatmen)m(t)f(of)g(the)g(op)s(eration)f Fr(M)59 b(7!)k Fm(e)-60 b Fu(\026)2930 4415 y Fq(k)2973 4400 y Fw(\()p Fr(M)p Fw(\))50 b(for)h(a)g(QP-)94 4516 y(represen)m(tation)42 b Fr(M)e Fw(=)h(\()p Fu(A;)17 b(S;)g(M)5 b(;)17 b(V)k Fw(\),)42 b(the)f(QP)f(\()p Fu(A;)17 b(S)6 b Fw(\))40 b(w)m(as)h(not)f(assumed)h(to)f(b)s(e)h(reduced.)94 4633 y(Recall)35 b(from)g(Prop)s(osition)1154 4633 y SDict begin H.S end 1154 4633 a Fw(10.5)1327 4568 y SDict begin H.R end 1327 4568 a 1327 4633 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.10.5) cvn H.B /ANN pdfmark end 1327 4633 a 36 w Fw(that)h(w)m(e)h(ha)m(v)m(e)g(a)f(w)m(ell-de\014ned)g(op)s (eration)f Fr(M)e(7!)g(M)3421 4648 y Fp(red)3558 4633 y Fw(on)94 4749 y(\(righ)m(t-equiv)-5 b(alence)37 b(classes)h(of)7 b(\))37 b(QP-represen)m(tations.)59 b(The)39 b(follo)m(wing)c(prop)s (ert)m(y)j(is)f(imme-)94 4865 y(diate)32 b(from)g(de\014nitions.)94 4865 y SDict begin H.S end 94 4865 a 94 4865 a SDict begin 14 H.A end 94 4865 a 94 4865 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.10.11) cvn H.B /DEST pdfmark end 94 4865 a 177 x FA(Prop)s(osition)d(10.11.)37 b Fv(L)-5 b(et)30 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\))28 b Fv(b)-5 b(e)29 b(a)g(QP)g(satisfying)g Fw(\()2295 5042 y SDict begin H.S end 2295 5042 a Fw(5.1)2420 4978 y SDict begin H.R end 2420 4978 a 2420 5042 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.1) cvn H.B /ANN pdfmark end 2420 5042 a Fw(\))g Fv(and)f(such)h(that)h(its)f(trivial)g(p)-5 b(art)94 5162 y Fw(\()p Fu(A)205 5177 y Fp(triv)322 5162 y Fu(;)17 b(S)432 5125 y Fp(\(2\))526 5162 y Fw(\))36 b Fv(\(se)-5 b(e)36 b Fw(\()839 5162 y SDict begin H.S end 839 5162 a Fw(4.4)963 5097 y SDict begin H.R end 963 5097 a 963 5162 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.4.4) cvn H.B /ANN pdfmark end 963 5162 a Fw(\))p Fv(\))g(has)g(no)f(arr) -5 b(ows)36 b(incident)f(to)h(a)g(vertex)g Fu(k)s Fv(.)49 b(Then,)35 b(for)h(every)g(r)-5 b(epr)g(e-)94 5278 y(sentation)35 b Fr(M)f Fv(of)h Fw(\()p Fu(A;)17 b(S)6 b Fw(\))p Fv(,)34 b(the)h(r)-5 b(epr)g(esentation)39 b Fm(e)-60 b Fu(\026)1966 5293 y Fq(k)2009 5278 y Fw(\()p Fr(M)p Fw(\))34 b Fv(is)h(right-e)-5 b(quivalent)34 b(to)39 b Fm(e)-59 b Fu(\026)3206 5293 y Fq(k)3248 5278 y Fw(\()p Fr(M)3406 5293 y Fp(red)3508 5278 y Fw(\))p Fv(.)194 5455 y Fw(Recall)27 b(that,)i(according)f(to)g (Corollary)1696 5455 y SDict begin H.S end 1696 5455 a Fw(5.4)1820 5391 y SDict begin H.R end 1820 5391 a 1820 5455 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.5.4) cvn H.B /ANN pdfmark end 1820 5455 a 29 w Fw(and)g(De\014nition)2484 5455 y SDict begin H.S end 2484 5455 a Fw(5.5)2609 5391 y SDict begin H.R end 2609 5391 a 2609 5455 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.5.5) cvn H.B /ANN pdfmark end 2609 5455 a Fw(,)h(the)g(corresp)s(ondence)35 b Fm(e)-60 b Fu(\026)3564 5470 y Fq(k)3634 5455 y Fw(:)94 5580 y(\()p Fu(A;)17 b(S)6 b Fw(\))32 b Fr(7!)37 b Fm(e)-60 b Fu(\026)576 5595 y Fq(k)619 5580 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\))32 b(=)g(\()1078 5555 y Fm(e)1056 5580 y Fu(A;)1186 5555 y Fm(e)1173 5580 y Fu(S)6 b Fw(\))35 b(giv)m(es)h(rise)f(to)g(the)h Fv(mutation)42 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\))32 b Fr(7!)g Fu(\026)2934 5595 y Fq(k)2977 5580 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\))32 b(=)g(\()p 3414 5500 74 4 v Fu(A;)p 3531 5500 66 4 v 17 w(S)5 b Fw(\),)94 5696 y(whic)m(h)27 b(is)e(a)h(w)m(ell-de\014ned)g (bijectiv)m(e)g(transformation)e(on)i(the)g(set)h(of)e(righ)m(t-equiv) -5 b(alence)25 b(classes)p eop end end %%Page: 47 47 TeXDict begin HPSdict begin 47 46 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.47) cvn H.B /DEST pdfmark end 94 218 a 1179 w Fs(QUIVERS)31 b(WITH)i(POTENTIALS)g(I)1102 b(47)94 431 y Fw(of)34 b(reduced)i(QPs)f(satisfying)f(\()1262 431 y SDict begin H.S end 1262 431 a Fw(5.1)1387 367 y SDict begin H.R end 1387 367 a 1387 431 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.1) cvn H.B /ANN pdfmark end 1387 431 a Fw(\).)48 b(Here)36 b(\()p 1771 351 74 4 v Fu(A;)p 1888 351 66 4 v 17 w(S)5 b Fw(\))34 b(is)g(the)h(reduced)h (part)e(of)g(\()3047 406 y Fm(e)3024 431 y Fu(A;)3155 406 y Fm(e)3141 431 y Fu(S)6 b Fw(\).)49 b(No)m(w)35 b(for)94 547 y(ev)m(ery)g(QP-represen)m(tation)e Fr(M)27 b Fw(=)h(\()p Fu(A;)17 b(S;)g(M)5 b(;)17 b(V)k Fw(\))33 b(of)f(a)g(reduced)i(QP)f(\()p Fu(A;)17 b(S)6 b Fw(\))32 b(w)m(e)i(de\014ne)94 710 y(\(10.22\))1413 710 y SDict begin H.S end 1413 710 a 1413 710 a SDict begin 14 H.A end 1413 710 a 1413 710 a SDict begin [ /View [/XYZ H.V] /Dest (equation.10.22) cvn H.B /DEST pdfmark end 1413 710 a Fu(\026)1472 725 y Fq(k)1514 710 y Fw(\()p Fr(M)p Fw(\))27 b(=)h(\()t Fm(e)-59 b Fu(\026)1938 725 y Fq(k)1980 710 y Fw(\()p Fr(M)p Fw(\)\))2214 725 y Fp(red)2316 710 y Fw(;)94 878 y(th)m(us,)52 b Fu(\026)413 893 y Fq(k)455 878 y Fw(\()p Fr(M)p Fw(\))46 b(is)g(a)g(decorated)i(represen)m(tation)f(\()p 2052 798 74 4 v Fu(A;)p 2169 798 66 4 v 17 w(S)6 b(;)p 2279 798 105 4 v 17 w(M)k(;)p 2427 798 79 4 v 17 w(V)21 b Fw(\))47 b(of)f(a)g(reduced)i(QP)f(\()p 3414 798 74 4 v Fu(A;)p 3531 798 66 4 v 17 w(S)5 b Fw(\).)94 994 y(Com)m(bining)27 b(Prop)s(ositions)1144 994 y SDict begin H.S end 1144 994 a Fw(10.5)1317 930 y SDict begin H.R end 1317 930 a 1317 994 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.10.5) cvn H.B /ANN pdfmark end 1317 994 a 28 w Fw(and)1530 994 y SDict begin H.S end 1530 994 a Fw(10.10)1752 930 y SDict begin H.R end 1752 930 a 1752 994 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.10.10) cvn H.B /ANN pdfmark end 1752 994 a Fw(,)i(w)m(e)g(obtain)d(the)i(follo)m (wing)d(imp)s(ortan)m(t)h(corollary)-8 b(.)94 1014 y SDict begin H.S end 94 1014 a 94 1014 a SDict begin 14 H.A end 94 1014 a 94 1014 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.10.12) cvn H.B /DEST pdfmark end 94 1014 a 158 x FA(Corollary)43 b(10.12.)h Fv(The)39 b(c)-5 b(orr)g(esp)g(ondenc)g(e) 37 b Fr(M)e(7!)g Fu(\026)2156 1187 y Fq(k)2199 1172 y Fw(\()p Fr(M)p Fw(\))j Fv(is)h(a)g(wel)5 b(l-de\014ne)-5 b(d)37 b(tr)-5 b(ansforma-)94 1288 y(tion)45 b(on)f(the)h(set)f(of)g (right-e)-5 b(quivalenc)g(e)43 b(classes)h(of)g(de)-5 b(c)g(or)g(ate)g(d)44 b(r)-5 b(epr)g(esentations)44 b(of)g(r)-5 b(e)g(duc)g(e)g(d)94 1404 y(QPs)35 b(satisfying)f Fw(\()781 1404 y SDict begin H.S end 781 1404 a Fw(5.1)906 1340 y SDict begin H.R end 906 1340 a 906 1404 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.1) cvn H.B /ANN pdfmark end 906 1404 a Fw(\))p Fv(.)194 1582 y Fw(W)-8 b(e)28 b(refer)f(to)f(the) i(transformation)d Fr(M)i(7!)g Fu(\026)1844 1597 y Fq(k)1887 1582 y Fw(\()p Fr(M)p Fw(\))f(in)g(Corollary)2643 1582 y SDict begin H.S end 2643 1582 a Fw(10.12)2865 1518 y SDict begin H.R end 2865 1518 a 2865 1582 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.10.12) cvn H.B /ANN pdfmark end 2865 1582 a 27 w Fw(as)h(the)g Fv(mutation)j(at)94 1698 y(vertex)38 b Fu(k)s Fw(.)k(With)25 b(some)h(abuse)h(of)f (terminology)-8 b(,)25 b(w)m(e)i(will)d(talk)h(ab)s(out)h(m)m(utations) f(of)h(decorated)94 1815 y(represen)m(tations)34 b(\(rather)f(than)f (their)h(righ)m(t-equiv)-5 b(alence)31 b(classes\).)194 1931 y(The)j(follo)m(wing)c(result)i(naturally)f(extends)k(Theorem)2265 1931 y SDict begin H.S end 2265 1931 a Fw(5.7)2390 1867 y SDict begin H.R end 2390 1867 a 2390 1931 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.5.7) cvn H.B /ANN pdfmark end 2390 1931 a Fw(.)94 1950 y SDict begin H.S end 94 1950 a 94 1950 a SDict begin 14 H.A end 94 1950 a 94 1950 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.10.13) cvn H.B /DEST pdfmark end 94 1950 a 159 x FA(Theorem)26 b(10.13.)35 b Fv(The)25 b(mutation)h Fu(\026)1535 2124 y Fq(k)1604 2109 y Fv(of)f(de)-5 b(c)g(or)g(ate)g(d)25 b(r)-5 b(epr)g(esentations) 26 b(is)f(an)h(involution;)i(that)94 2225 y(is,)33 b(for)f(every)h(de) -5 b(c)g(or)g(ate)g(d)32 b(r)-5 b(epr)g(esentation)32 b Fr(M)g Fv(of)g(a)g(r)-5 b(e)g(duc)g(e)g(d)33 b(QP)f Fw(\()p Fu(A;)17 b(S)6 b Fw(\))32 b Fv(satisfying)g Fw(\()3309 2225 y SDict begin H.S end 3309 2225 a Fw(5.1)3434 2161 y SDict begin H.R end 3434 2161 a 3434 2225 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.1) cvn H.B /ANN pdfmark end 3434 2225 a Fw(\))p Fv(,)g(the)94 2341 y(de)-5 b(c)g(or)g(ate)g(d)35 b(r)-5 b(epr)g(esentation)34 b Fu(\026)1211 2305 y Fp(2)1211 2367 y Fq(k)1253 2341 y Fw(\()p Fr(M)p Fw(\))h Fv(of)f(a)h(QP)f Fu(\026)1917 2305 y Fp(2)1917 2367 y Fq(k)1960 2341 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\))34 b Fv(is)h(right-e)-5 b(quivalent)34 b(to)h Fr(M)p Fv(.)94 2519 y(Pr)-5 b(o)g(of.)42 b Fw(In)25 b(view)g(of)g(Prop)s(osition)1345 2519 y SDict begin H.S end 1345 2519 a Fw(10.11)1567 2455 y SDict begin H.R end 1567 2455 a 1567 2519 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.10.11) cvn H.B /ANN pdfmark end 1567 2519 a Fw(,)i Fu(\026)1680 2483 y Fp(2)1680 2545 y Fq(k)1722 2519 y Fw(\()p Fr(M)p Fw(\))e(is)f(righ)m(t-equiv)-5 b(alen)m(t)24 b(to)h(\()t Fm(e)-59 b Fu(\026)2934 2483 y Fp(2)2934 2545 y Fq(k)2976 2519 y Fw(\()p Fr(M)p Fw(\)\))3210 2534 y Fp(red)3312 2519 y Fw(.)41 b(There-)94 2635 y(fore,)33 b(is)f(su\016ces)j(to)d(sho)m(w)i(that)e(\()t Fm(e)-59 b Fu(\026)1413 2599 y Fp(2)1413 2661 y Fq(k)1455 2635 y Fw(\()p Fr(M)p Fw(\)\))1689 2650 y Fp(red)1824 2635 y Fw(is)32 b(righ)m(t-equiv)-5 b(alen)m(t)31 b(to)h Fr(M)p Fw(.)194 2796 y(W)-8 b(e)28 b(write)g(the)g(QP-represen)m(tation)k Fm(e)-59 b Fu(\026)1631 2759 y Fp(2)1631 2821 y Fq(k)1673 2796 y Fw(\()p Fr(M)p Fw(\))27 b(as)33 b Fm(e)-60 b Fu(\026)2070 2759 y Fp(2)2070 2821 y Fq(k)2113 2796 y Fw(\()p Fr(M)p Fw(\))27 b(=)g(\()2486 2741 y Fm(e)2500 2770 y(e)2477 2796 y Fu(A;)2599 2741 y Fm(e)2607 2770 y(e)2594 2796 y Fu(S)6 b(;)p 2704 2696 105 4 v 2704 2715 V 17 w(M)k(;)p 2852 2696 79 4 v 2852 2715 V 17 w(V)22 b Fw(\).)41 b(The)29 b(QP)f(\()3450 2741 y Fm(e)3463 2770 y(e)3441 2796 y Fu(A;)3563 2741 y Fm(e)3571 2770 y(e)3558 2796 y Fu(S)5 b Fw(\))94 2912 y(is)35 b(giv)m(en)g(b)m(y)h(\()628 2912 y SDict begin H.S end 628 2912 a Fw(5.18)802 2848 y SDict begin H.R end 802 2848 a 802 2912 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.18) cvn H.B /ANN pdfmark end 802 2912 a Fw(\))f(and)g(\()1105 2912 y SDict begin H.S end 1105 2912 a Fw(5.19)1278 2848 y SDict begin H.R end 1278 2848 a 1278 2912 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.19) cvn H.B /ANN pdfmark end 1278 2912 a Fw(\).)51 b(In)36 b(particular,)e(there)h(is)g(a)g(natural)f(em)m(b)s(edding)g (of)h Fu(A)g Fw(in)m(to)103 3017 y Fm(e)117 3046 y(e)94 3071 y Fu(A)g Fw(iden)m(tifying)f Fu(A)h Fw(with)f(the)h(reduced)i (part)1780 3017 y Fm(e)1794 3046 y(e)1772 3071 y Fu(A)1845 3086 y Fp(red)1947 3071 y Fw(.)50 b(F)-8 b(urthermore,)35 b(as)g(sho)m(wn)h(in)f(the)g(pro)s(of)f(of)94 3225 y(Theorem)511 3225 y SDict begin H.S end 511 3225 a Fw(5.7)636 3161 y SDict begin H.R end 636 3161 a 636 3225 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.5.7) cvn H.B /ANN pdfmark end 636 3225 a Fw(,)k(an)f(automorphism)e(of)i Fu(R)q Fr(hh)1769 3171 y Fm(e)1782 3200 y(e)1760 3225 y Fu(A)p Fr(ii)g Fw(that)g(establishes)g(the)h(righ)m(t-equiv)-5 b(alence)36 b(in)94 3341 y(\()132 3341 y SDict begin H.S end 132 3341 a Fw(5.17)306 3277 y SDict begin H.R end 306 3277 a 306 3341 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.17) cvn H.B /ANN pdfmark end 306 3341 a Fw(\))d(can)g(b)s(e)h(c)m(hosen)g (so)g(that)f(it)f(restricts)h(to)g(an)g(automorphism)e Fu(')2774 3356 y Fp(0)2842 3341 y Fw(:)e Fu(R)q Fr(hh)p Fu(A)p Fr(ii)f(!)g Fu(R)q Fr(hh)p Fu(A)p Fr(ii)94 3457 y Fw(acting)k(as)h(follo)m(ws:)835 3620 y Fu(')899 3635 y Fp(0)971 3620 y Fw(m)m(ultiplies)c(eac)m(h)34 b(of)e(the)h(arro)m(ws) g Fu(b)2270 3635 y Fp(1)2310 3620 y Fu(;)17 b(:)g(:)g(:)f(;)h(b)2570 3635 y Fq(t)2632 3620 y Fw(b)m(y)34 b Fr(\000)p Fw(1,)94 3620 y SDict begin H.S end 94 3620 a 94 3620 a SDict begin 17.5 H.A end 94 3620 a 94 3620 a SDict begin [ /View [/XYZ H.V] /Dest (equation.10.23) cvn H.B /DEST pdfmark end 94 3620 a Fw(\(10.23\))835 3766 y(and)f(\014xes)h(the)f(rest)g(of)f(the)h(arro) m(ws)g(in)f Fu(A)p Fw(.)94 3939 y(In)i(view)f(of)g(De\014nition)1008 3939 y SDict begin H.S end 1008 3939 a Fw(10.4)1181 3875 y SDict begin H.R end 1181 3875 a 1181 3939 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.10.4) cvn H.B /ANN pdfmark end 1181 3939 a Fw(,)g(the)h(QP-represen)m(tation)f Fu(\026)2281 3903 y Fp(2)2281 3965 y Fq(k)2323 3939 y Fw(\()p Fr(M)p Fw(\))28 b(=)h(\()t Fm(e)-59 b Fu(\026)2749 3903 y Fp(2)2749 3965 y Fq(k)2791 3939 y Fw(\()p Fr(M)p Fw(\)\))3025 3954 y Fp(red)3159 3939 y Fw(can)34 b(b)s(e)f(real-)94 4081 y(ized)i(as)h(\()p Fu(A;)17 b(S;)g(M)783 4045 y Fn(0)807 4081 y Fu(;)g(V)929 4045 y Fn(0)952 4081 y Fw(\),)36 b(where)g Fu(M)1441 4045 y Fn(0)1497 4081 y Fw(=)p 1605 3981 105 4 v 1605 4001 V 32 w Fu(M)46 b Fw(and)35 b Fu(V)2015 4045 y Fn(0)2071 4081 y Fw(=)p 2178 3981 79 4 v 2178 4001 V 31 w Fu(V)57 b Fw(as)35 b(v)m(ector)h(spaces,)i(and)d(the)g (action)94 4197 y(of)e Fu(R)q Fr(hh)p Fu(A)p Fr(ii)e Fw(in)h Fu(M)759 4161 y Fn(0)816 4197 y Fw(is)g(giv)m(en)g(b)m(y)94 4360 y(\(10.24\))1188 4360 y SDict begin H.S end 1188 4360 a 1188 4360 a SDict begin 14 H.A end 1188 4360 a 1188 4360 a SDict begin [ /View [/XYZ H.V] /Dest (equation.10.24) cvn H.B /DEST pdfmark end 1188 4360 a Fu(u)1244 4375 y Fq(M)1319 4356 y Ff(0)1372 4360 y Fw(=)c Fu(')1540 4375 y Fp(0)1579 4360 y Fw(\()p Fu(u)p Fw(\))p 1711 4332 79 3 v 1711 4347 75 3 v 41 x Fq(M)1985 4360 y Fw(\()p Fu(u)f Fr(2)h Fu(R)q Fr(hh)p Fu(A)p Fr(ii)p Fw(\))p Fu(:)94 4531 y Fw(T)-8 b(o)22 b(pro)m(v)m(e)h(Theorem)880 4531 y SDict begin H.S end 880 4531 a Fw(10.13)1103 4466 y SDict begin H.R end 1103 4466 a 1103 4531 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.10.13) cvn H.B /ANN pdfmark end 1103 4531 a Fw(,)g(it)e(su\016ces)j(to)d(sho)m(w)i(that)f(the)g (decorated)g(represen)m(tation)h(\()p Fu(M)3460 4494 y Fn(0)3484 4531 y Fu(;)17 b(V)3606 4494 y Fn(0)3630 4531 y Fw(\))94 4647 y(of)33 b(\()p Fu(A;)17 b(S)6 b Fw(\))32 b(is)g(isomorphic)e(to)j(\()p Fu(M)5 b(;)17 b(V)k Fw(\).)194 4788 y(W)-8 b(e)34 b(\014rst)g(compute)g Fu(M)1066 4752 y Fn(0)1119 4788 y Fw(=)p 1224 4688 105 4 v 1224 4708 V 29 w Fu(M)44 b Fw(and)34 b Fu(V)1632 4752 y Fn(0)1684 4788 y Fw(=)p 1789 4688 79 4 v 1789 4708 V 29 w Fu(V)55 b Fw(as)34 b(v)m(ector)h(spaces.)48 b(According)33 b(to)g(\()3278 4788 y SDict begin H.S end 3278 4788 a Fw(10.6)3452 4724 y SDict begin H.R end 3452 4724 a 3452 4788 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.6) cvn H.B /ANN pdfmark end 3452 4788 a Fw(\),)g(w)m(e)94 4904 y(ha)m(v)m(e)955 5026 y Fu(M)1059 4985 y Fn(0)1049 5051 y Fq(i)1111 5026 y Fw(=)p 1215 4926 105 4 v 1215 4946 V 28 w Fu(M)1319 5041 y Fq(i)1375 5026 y Fw(=)p 1479 4946 V 28 w Fu(M)1583 5041 y Fq(i)1639 5026 y Fw(=)28 b Fu(M)1837 5041 y Fq(i)1865 5026 y Fu(;)115 b(V)2085 4985 y Fn(0)2064 5051 y Fq(i)2136 5026 y Fw(=)p 2240 4926 79 4 v 2240 4946 V 28 w Fu(V)2318 5041 y Fq(i)2374 5026 y Fw(=)p 2478 4946 V 28 w Fu(V)2556 5041 y Fq(i)2612 5026 y Fw(=)28 b Fu(V)2773 5041 y Fq(i)94 5169 y Fw(for)g(all)e Fu(i)i Fr(6)p Fw(=)f Fu(k)s Fw(.)42 b(As)28 b(for)g(the)g(spaces)h Fu(M)1502 5133 y Fn(0)1492 5195 y Fq(k)1563 5169 y Fw(and)f Fu(V)1827 5133 y Fn(0)1805 5195 y Fq(k)1850 5169 y Fw(,)h(they)g(are)f (giv)m(en)g(as)g(in)f(\()2791 5169 y SDict begin H.S end 2791 5169 a Fw(10.7)2964 5105 y SDict begin H.R end 2964 5105 a 2964 5169 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.7) cvn H.B /ANN pdfmark end 2964 5169 a Fw(\),)i(with)e(the)i(maps) 94 5292 y Fu(\013)q(;)17 b(\014)6 b Fw(,)35 b(and)g Fu(\015)40 b Fw(replaced)35 b(b)m(y)p 1132 5237 63 4 v 36 w Fu(\013)q(;)p 1239 5211 61 4 v 17 w(\014)5 b Fw(,)36 b(and)p 1554 5237 57 4 v 35 w Fu(\015)5 b Fw(,)35 b(resp)s(ectiv)m(ely)-8 b(.)52 b(Recall)33 b(that)p 2751 5237 63 4 v 35 w Fu(\013)j Fw(and)p 3041 5211 61 4 v 35 w Fu(\014)k Fw(are)35 b(giv)m(en)g(b)m(y) 94 5417 y(\()132 5417 y SDict begin H.S end 132 5417 a Fw(10.10)355 5353 y SDict begin H.R end 355 5353 a 355 5417 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.10) cvn H.B /ANN pdfmark end 355 5417 a Fw(\).)43 b(As)32 b(for)p 754 5362 57 4 v 31 w Fu(\015)5 b Fw(,)31 b(b)m(y)i(applying)d (the)i(de\014nition)e(\()2037 5417 y SDict begin H.S end 2037 5417 a Fw(10.4)2211 5353 y SDict begin H.R end 2211 5353 a 2211 5417 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.4) cvn H.B /ANN pdfmark end 2211 5417 a Fw(\))h(to)g(the)h(p)s(oten) m(tial)2989 5392 y Fm(e)2976 5417 y Fu(S)37 b Fw(giv)m(en)32 b(b)m(y)g(\()3499 5417 y SDict begin H.S end 3499 5417 a Fw(5.8)3623 5353 y SDict begin H.R end 3623 5353 a 3623 5417 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.8) cvn H.B /ANN pdfmark end 3623 5417 a Fw(\))94 5533 y(and)h(\()322 5533 y SDict begin H.S end 322 5533 a Fw(5.9)447 5469 y SDict begin H.R end 447 5469 a 447 5533 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.9) cvn H.B /ANN pdfmark end 447 5533 a Fw(\),)f(w)m(e)i(see)g(that)94 5696 y(\(10.25\))1709 5696 y SDict begin H.S end 1709 5696 a 1709 5696 a SDict begin 14 H.A end 1709 5696 a 1709 5696 a SDict begin [ /View [/XYZ H.V] /Dest (equation.10.25) cvn H.B /DEST pdfmark end 1709 5696 a 1709 5641 57 4 v Fu(\015)f Fw(=)27 b Fu(\014)6 b(\013)q(:)p eop end end %%Page: 48 48 TeXDict begin HPSdict begin 48 47 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.48) cvn H.B /DEST pdfmark end 94 218 a Fs(48)492 b(HARM)25 b(DERKSEN,)f(JERZY)i(WEYMAN,)f(AND)f(ANDREI)g(ZELEVINSKY)94 417 y Fw(As)34 b(a)e(direct)g(consequence)k(of)c(the)h(de\014nitions,)f (w)m(e)i(conclude)f(that)748 580 y SDict begin H.S end 748 580 a 748 580 a SDict begin 14 H.A end 748 580 a 748 580 a SDict begin [ /View [/XYZ H.V] /Dest (equation.10.26) cvn H.B /DEST pdfmark end 748 580 a 83 w Fw(k)m(er)p 977 525 63 4 v 17 w Fu(\013)c Fw(=)e(im)k Fu(\014)6 b(;)114 b Fw(im)p 1655 525 V 31 w Fu(\013)28 b Fw(=)1863 536 y Fp(k)n(er)11 b Fq(\015)p 1859 557 154 4 v 1859 614 a Fp(im)31 b Fq(\014)2045 580 y Fr(\010)22 b Fw(im)31 b Fu(\015)c Fr(\010)c(f)p Fw(0)p Fr(g)f(\010)g(f)p Fw(0)p Fr(g)p Fu(;)853 754 y Fw(k)m(er)p 999 673 61 4 v 17 w Fu(\014)33 b Fw(=)1205 710 y Fp(k)n(er)12 b Fq(\015)p 1201 731 154 4 v 1201 789 a Fp(im)31 b Fq(\014)1387 754 y Fr(\010)23 b(f)p Fw(0)p Fr(g)e(\010)i(f)p Fw(0)p Fr(g)e(\010)i Fu(V)2084 769 y Fq(k)2127 754 y Fu(;)164 b Fw(im)p 2458 673 61 4 v 30 w Fu(\014)34 b Fw(=)27 b(k)m(er)18 b Fu(\013)q(;)-2793 b Fw(\(10.26\))1128 914 y(k)m(er)p 1275 859 57 4 v 18 w Fu(\015)32 b Fw(=)c(k)m(er)q(\()p Fu(\014)6 b(\013)q Fw(\))p Fu(;)113 b Fw(im)p 2073 859 V 31 w Fu(\015)33 b Fw(=)28 b(im)j(\()p Fu(\014)6 b(\013)q Fw(\))p Fu(:)94 1075 y Fw(It)33 b(follo)m(ws)e(that)1236 1244 y Fu(V)1314 1203 y Fn(0)1293 1269 y Fq(k)1365 1244 y Fw(=)p 1469 1144 79 4 v 1469 1164 V 28 w Fu(V)1547 1259 y Fq(k)1617 1244 y Fw(=)1888 1177 y(k)m(er)p 2035 1096 61 4 v 18 w Fu(\014)p 1731 1221 522 4 v 1731 1322 a Fw(k)m(er)p 1878 1241 61 4 v 18 w Fu(\014)c Fr(\\)c Fw(im)p 2190 1268 63 4 v 31 w Fu(\013)2290 1244 y Fw(=)28 b Fu(V)2451 1259 y Fq(k)2493 1244 y Fu(;)94 1448 y Fw(and)33 b(so)g Fu(V)482 1412 y Fn(0)533 1448 y Fw(=)28 b Fu(V)22 b Fw(,)32 b(as)h(desired.)44 b(W)-8 b(e)33 b(also)f(ha)m(v)m(e)736 1663 y Fu(M)840 1622 y Fn(0)830 1688 y Fq(k)901 1663 y Fw(=)1014 1596 y(k)m(er)q(\()p Fu(\014)6 b(\013)q Fw(\))p 1014 1640 330 4 v 1074 1731 a(k)m(er)18 b Fu(\013)1376 1663 y Fr(\010)k Fw(im)31 b(\()p Fu(\014)6 b(\013)q Fw(\))21 b Fr(\010)2016 1596 y Fw(im)31 b Fu(\014)p 1947 1640 341 4 v 1947 1731 a Fw(im)g(\()p Fu(\014)6 b(\013)q Fw(\))2319 1663 y Fr(\010)2586 1596 y Fw(k)m(er)18 b Fu(\014)p 2429 1640 522 4 v 2429 1731 a Fw(k)m(er)f Fu(\014)28 b Fr(\\)23 b Fw(im)31 b Fu(\013)2993 1663 y(:)94 1879 y Fw(W)-8 b(e)33 b(no)m(w)h(mak)m(e)e(the)h(follo)m(wing)d(easy)k(observ)-5 b(ations:)318 2018 y Fr(\017)41 b Fw(the)33 b(map)f Fu(\013)h Fw(induces)g(an)g(isomorphism)d(k)m(er)q(\()p Fu(\014)6 b(\013)q Fw(\))p Fu(=)17 b Fw(k)m(er)f Fu(\013)29 b Fr(!)e Fw(k)m(er)18 b Fu(\014)27 b Fr(\\)c Fw(im)31 b Fu(\013)q Fw(;)318 2134 y Fr(\017)41 b Fw(the)33 b(map)f Fu(\014)38 b Fw(induces)33 b(an)g(isomorphism)d(im)h Fu(\013)q(=)p Fw(\(k)m(er)17 b Fu(\014)28 b Fr(\\)22 b Fw(im)31 b Fu(\013)q Fw(\))c Fr(!)h Fw(im)j(\()p Fu(\014)6 b(\013)q Fw(\);)318 2250 y Fr(\017)41 b Fw(the)33 b(map)f Fu(\014)38 b Fw(induces)33 b(an)g(isomorphism)d Fu(M)2034 2265 y Fq(k)2077 2250 y Fu(=)p Fw(\(k)m(er)17 b Fu(\014)28 b Fw(+)22 b(im)31 b Fu(\013)q Fw(\))c Fr(!)h Fw(im)i Fu(\014)6 b(=)p Fw(im)31 b(\()p Fu(\014)6 b(\013)q Fw(\).)318 2366 y Fr(\017)41 b Fw(there)33 b(is)f(a)h(natural)e(isomorphism)f(k)m(er)18 b Fu(\014)6 b(=)p Fw(\(k)m(er)17 b Fu(\014)28 b Fr(\\)22 b Fw(im)31 b Fu(\013)q Fw(\))c Fr(!)h Fw(\(k)m(er)17 b Fu(\014)28 b Fw(+)22 b(im)31 b Fu(\013)q Fw(\))p Fu(=)p Fw(im)f Fu(\013)q Fw(.)94 2505 y(Using)j(these)h(isomorphisms,)c(w)m(e) k(can)f(iden)m(tify)f Fu(M)2036 2469 y Fn(0)2026 2531 y Fq(k)2101 2505 y Fw(with)g(the)h(space)94 2721 y(\(10.27\))560 2721 y SDict begin H.S end 560 2721 a 560 2721 a SDict begin 14 H.A end 560 2721 a 560 2721 a SDict begin [ /View [/XYZ H.V] /Dest (equation.10.27) cvn H.B /DEST pdfmark end 560 2721 a Fu(M)664 2680 y Fn(0)654 2746 y Fq(k)725 2721 y Fw(=)27 b(\(k)m(er)18 b Fu(\014)28 b Fr(\\)22 b Fw(im)31 b Fu(\013)q Fw(\))22 b Fr(\010)1716 2654 y Fw(im)31 b Fu(\013)p 1557 2698 522 4 v 1557 2789 a Fw(k)m(er)18 b Fu(\014)28 b Fr(\\)22 b Fw(im)31 b Fu(\013)2111 2721 y Fr(\010)2418 2654 y Fu(M)2512 2669 y Fq(k)p 2221 2698 532 4 v 2221 2789 a Fw(k)m(er)17 b Fu(\014)28 b Fw(+)22 b(im)31 b Fu(\013)2784 2721 y Fr(\010)2893 2654 y Fw(k)m(er)18 b Fu(\014)28 b Fw(+)22 b(im)31 b Fu(\013)p 2893 2698 V 3057 2789 a Fw(im)g Fu(\013)3467 2721 y(:)194 2937 y Fw(T)-8 b(o)38 b(describ)s(e)h(the)f(action)f(of)h Fu(R)q Fr(hh)p Fu(A)p Fr(ii)f Fw(in)g Fu(M)1878 2901 y Fn(0)1902 2937 y Fw(,)j(w)m(e)f(need)g(only)e(to)h(describ)s(e)g(the) h(maps)e Fu(\013)3574 2901 y Fn(0)3634 2937 y Fw(:)94 3053 y Fu(M)188 3068 y Fp(in)292 3053 y Fr(!)i Fu(M)535 3017 y Fn(0)525 3079 y Fq(k)608 3053 y Fw(and)h Fu(\014)866 3017 y Fn(0)929 3053 y Fw(:)g Fu(M)1100 3017 y Fn(0)1090 3079 y Fq(k)1173 3053 y Fr(!)g Fu(M)1407 3068 y Fp(out)1553 3053 y Fw(constructed)h(in)e(the)i(same)e(w)m(a)m(y)i(as)f(in)f(\()3125 3053 y SDict begin H.S end 3125 3053 a Fw(10.3)3299 2989 y SDict begin H.R end 3299 2989 a 3299 3053 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.3) cvn H.B /ANN pdfmark end 3299 3053 a Fw(\).)65 b(As)40 b(in)94 3169 y(\()132 3169 y SDict begin H.S end 132 3169 a Fw(10.10)355 3105 y SDict begin H.R end 355 3105 a 355 3169 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.10) cvn H.B /ANN pdfmark end 355 3169 a Fw(\),)33 b(the)g(de\014nition)f(of)g Fu(\013)1229 3133 y Fn(0)1285 3169 y Fw(and)h Fu(\014)1536 3133 y Fn(0)1592 3169 y Fw(in)m(v)m(olv)m(es)h(splitting)c(data)j(\() 2605 3169 y SDict begin H.S end 2605 3169 a Fw(10.8)2778 3105 y SDict begin H.R end 2778 3105 a 2778 3169 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.8) cvn H.B /ANN pdfmark end 2778 3169 a Fw(\))g(-)f(\()2952 3169 y SDict begin H.S end 2952 3169 a Fw(10.9)3126 3105 y SDict begin H.R end 3126 3105 a 3126 3169 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.9) cvn H.B /ANN pdfmark end 3126 3169 a Fw(\).)44 b(In)34 b(view)f(of)94 3286 y(\()132 3286 y SDict begin H.S end 132 3286 a Fw(10.26)355 3221 y SDict begin H.R end 355 3221 a 355 3286 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.26) cvn H.B /ANN pdfmark end 355 3286 a Fw(\),)f(in)g(the)h(curren)m(t)h(situation)d(the)i (splitting)d(data)i(tak)m(e)i(the)f(follo)m(wing)c(form:)245 3447 y(Cho)s(ose)k(a)g(linear)e(map)p 1153 3392 51 4 v 31 w Fu(\032)d Fw(:)g Fu(M)1380 3462 y Fp(in)1471 3447 y Fr(!)g Fw(k)m(er)q(\()p Fu(\014)6 b(\013)q Fw(\))31 b(suc)m(h)j(that)p 2392 3392 V 33 w Fu(\032\023)28 b Fw(=)g(id)2689 3463 y Fp(k)n(er\()p Fq(\014)s(\013)p Fp(\))2930 3447 y Fw(.)245 3598 y(Cho)s(ose)33 b(a)g(linear)e(map)p 1153 3543 60 4 v 31 w Fu(\033)h Fw(:)c(im)j Fu(\014)6 b(=)p Fw(im)30 b(\()p Fu(\014)6 b(\013)q Fw(\))27 b Fr(!)g Fw(im)k Fu(\014)38 b Fw(suc)m(h)c(that)f Fu(\031)p 2765 3543 V 4 w(\033)e Fw(=)d(id)3036 3614 y Fp(im)j Fq(\014)s(=)p Fp(im)g(\()p Fq(\014)s(\013)p Fp(\))3484 3598 y Fw(.)94 3765 y(Adopting)g(\()562 3765 y SDict begin H.S end 562 3765 a Fw(10.10)784 3701 y SDict begin H.R end 784 3701 a 784 3765 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.10) cvn H.B /ANN pdfmark end 784 3765 a Fw(\))h(to)f(the)h(curren)m(t)h (situation)c(\(in)i(particular,)f(realizing)f Fu(M)3002 3729 y Fn(0)2992 3791 y Fq(k)3067 3765 y Fw(as)j(in)e(\()3336 3765 y SDict begin H.S end 3336 3765 a Fw(10.27)3558 3701 y SDict begin H.R end 3558 3701 a 3558 3765 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.27) cvn H.B /ANN pdfmark end 3558 3765 a Fw(\)\),)94 3882 y(w)m(e)k(see)g(that)e(the)h(maps)f Fu(\013)1093 3845 y Fn(0)1149 3882 y Fw(and)h Fu(\014)1400 3845 y Fn(0)1455 3882 y Fw(tak)m(e)g(the)g(follo)m(wing)d(form:)94 4218 y(\(10.28\))971 4218 y SDict begin H.S end 971 4218 a 971 4218 a SDict begin 14 H.A end 971 4218 a 971 4218 a SDict begin [ /View [/XYZ H.V] /Dest (equation.10.28) cvn H.B /DEST pdfmark end 971 4218 a Fu(\013)1034 4177 y Fn(0)1084 4218 y Fw(=)1188 3958 y Fm(0)1188 4134 y(B)1188 4194 y(B)1188 4257 y(@)1279 4042 y Fr(\000)p Fu(\013)p 1419 3988 51 4 v 1 w(\032)1275 4159 y Fr(\000)p Fu(\031)t(\013)1350 4275 y Fw(0)1350 4391 y(0)1474 3958 y Fm(1)1474 4134 y(C)1474 4194 y(C)1474 4257 y(A)1578 4218 y Fu(;)114 b(\014)1780 4177 y Fn(0)1831 4218 y Fw(=)1934 4138 y Fm(\000)1980 4216 y Fw(0)83 b Fr(\000)p Fu(\014)89 b Fr(\000)p Fu(\023)p 2444 4162 60 4 v 1 x(\033)2504 4216 y(\014)f Fw(0)2696 4138 y Fm(\001)2758 4218 y Fw(;)94 4555 y(here)41 b(with)e(some)h(abuse)h(of)e(notation)g(w)m(e)h(denote)h (b)m(y)g(the)f(same)g(sym)m(b)s(ol)f Fu(\014)45 b Fw(the)40 b(t)m(w)m(o)h(maps)94 4672 y(im)34 b Fu(\013)q(=)17 b Fw(k)m(er)f Fu(\014)29 b Fr(\\)24 b Fw(im)34 b Fu(\013)e Fr(!)f Fu(M)1149 4687 y Fp(out)1290 4672 y Fw(and)j Fu(M)1575 4687 y Fq(k)1619 4672 y Fu(=)p Fw(\(k)m(er)17 b Fu(\014)29 b Fw(+)24 b(im)33 b Fu(\013)q Fw(\))e Fr(!)g Fw(im)i Fu(\014)6 b(=)p Fw(im)32 b(\()p Fu(\014)6 b(\013)q Fw(\))34 b(induced)h(b)m(y)h Fu(\014)6 b Fw(.)94 4788 y(Note)43 b(that)g(the)g(app)s(earance)g(of)f(the)h(min)m(us)f(sign)g(in)g Fu(\014)2252 4752 y Fn(0)2317 4788 y Fw(is)h(caused)g(b)m(y)h(the)f (min)m(us)f(sign)g(in)94 4904 y(\()132 4904 y SDict begin H.S end 132 4904 a Fw(10.23)355 4840 y SDict begin H.R end 355 4840 a 355 4904 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.23) cvn H.B /ANN pdfmark end 355 4904 a Fw(\).)194 5020 y(T)-8 b(o)33 b(complete)f(the)h(pro)s(of)f(of)g (Theorem)1696 5020 y SDict begin H.S end 1696 5020 a Fw(10.13)1919 4956 y SDict begin H.R end 1919 4956 a 1919 5020 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.10.13) cvn H.B /ANN pdfmark end 1919 5020 a Fw(,)g(it)g(remains)g(to)g (construct)i(an)f(isomorphism)94 5136 y(of)g(v)m(ector)g(spaces)h Fu( )e Fw(:)c Fu(M)1051 5100 y Fn(0)1041 5162 y Fq(k)1112 5136 y Fr(!)f Fu(M)1333 5151 y Fq(k)1408 5136 y Fw(suc)m(h)34 b(that)94 5298 y(\(10.29\))1400 5298 y SDict begin H.S end 1400 5298 a 1400 5298 a SDict begin 14 H.A end 1400 5298 a 1400 5298 a SDict begin [ /View [/XYZ H.V] /Dest (equation.10.29) cvn H.B /DEST pdfmark end 1400 5298 a Fu( )t(\013)1530 5257 y Fn(0)1581 5298 y Fw(=)27 b Fu(\013)q(;)212 b(\014)6 b( )31 b Fw(=)d Fu(\014)2306 5257 y Fn(0)2329 5298 y Fu(:)94 5459 y Fw(T)-8 b(o)40 b(do)f(this,)h(notice)f(that)g(the)g (four)g(direct)g(summands)g(in)f(\()2439 5459 y SDict begin H.S end 2439 5459 a Fw(10.27)2661 5395 y SDict begin H.R end 2661 5395 a 2661 5459 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.27) cvn H.B /ANN pdfmark end 2661 5459 a Fw(\))h(are)g(the)h(factors)f(in)f(the)94 5575 y(\014ltration)828 5696 y Fr(f)p Fw(0)p Fr(g)27 b(\022)h Fw(k)m(er)18 b Fu(\014)28 b Fr(\\)22 b Fw(im)31 b Fu(\013)d Fr(\022)h Fw(im)h Fu(\013)f Fr(\022)f Fw(k)m(er)18 b Fu(\014)27 b Fw(+)22 b(im)31 b Fu(\013)e Fr(\022)f Fu(M)2858 5711 y Fq(k)2901 5696 y Fu(:)p eop end end %%Page: 49 49 TeXDict begin HPSdict begin 49 48 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.49) cvn H.B /DEST pdfmark end 94 218 a 1179 w Fs(QUIVERS)31 b(WITH)i(POTENTIALS)g(I)1102 b(49)94 417 y Fw(Cho)s(ose)34 b(some)e(sections)965 574 y Fu(\033)1020 589 y Fp(1)1088 574 y Fw(:)c(im)i Fu(\013)q(=)p Fw(\(k)m(er)17 b Fu(\014)28 b Fr(\\)23 b Fw(im)31 b Fu(\013)q Fw(\))c Fr(!)g Fw(im)k Fu(\013)q(;)965 725 y(\033)1020 740 y Fp(2)1088 725 y Fw(:)d(\(k)m(er)17 b Fu(\014)28 b Fw(+)22 b(im)31 b Fu(\013)q Fw(\))p Fu(=)p Fw(im)f Fu(\013)e Fr(!)g Fw(\(k)m(er)17 b Fu(\014)28 b Fw(+)22 b(im)31 b Fu(\013)q Fw(\))p Fu(;)965 876 y(\033)1020 891 y Fp(3)1088 876 y Fw(:)d Fu(M)1237 891 y Fq(k)1280 876 y Fu(=)p Fw(\(k)m(er)17 b Fu(\014)28 b Fw(+)22 b(im)31 b Fu(\013)q Fw(\))c Fr(!)g Fu(M)2184 891 y Fq(k)94 1033 y Fw(for)33 b(the)g(three)g(factors)f(of)g (this)h(\014ltration,)d(so)j(that)f(they)i(satisfy:)635 1190 y(im)d Fu(\033)831 1205 y Fp(1)898 1190 y Fw(=)d Fu(\013)q Fw(\(k)m(er)p 1249 1135 51 4 v 17 w Fu(\032)p Fw(\))p Fu(;)212 b Fw(im)31 b Fu(\033)1772 1205 y Fp(2)1839 1190 y Fr(\022)e Fw(k)m(er)17 b Fu(\014)6 b(;)212 b Fw(im)31 b(\()p Fu(\014)6 b(\033)2686 1205 y Fp(3)2725 1190 y Fw(\))28 b(=)f(im)p 3035 1135 60 4 v 31 w Fu(\033)t(:)94 1347 y Fw(No)m(w)34 b(de\014ne)f(an)g(an)f(isomorphism)f Fu( )g Fw(:)d Fu(M)1693 1311 y Fn(0)1683 1373 y Fq(k)1754 1347 y Fr(!)f Fu(M)1975 1362 y Fq(k)2051 1347 y Fw(b)m(y)33 b(setting)1221 1507 y Fu( )e Fw(=)1419 1426 y Fm(\000)1464 1505 y Fr(\000)p Fu(\023)84 b Fr(\000)p Fu(\023\033)1825 1520 y Fp(1)1949 1505 y Fr(\000)p Fu(\023\033)2115 1520 y Fp(3)2239 1505 y Fr(\000)p Fu(\023\033)2405 1520 y Fp(2)2446 1426 y Fm(\001)2508 1507 y Fu(:)94 1665 y Fw(The)34 b(equalities)d(\()764 1665 y SDict begin H.S end 764 1665 a Fw(10.29)987 1601 y SDict begin H.R end 987 1601 a 987 1665 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.29) cvn H.B /ANN pdfmark end 987 1665 a Fw(\))h(are)h(c)m(hec)m(k)m(ed) i(b)m(y)f(a)e(direct)g(insp)s(ection,)g(\014nishing)g(the)h(pro)s(of.) 214 b Fh(\003)194 1831 y Fw(Note)50 b(that)f(there)i(is)e(an)g(ob)m (vious)h(w)m(a)m(y)h(to)e(de\014ne)i(direct)f(sums)g(for)f(decorated)h (repre-)94 1948 y(sen)m(tations)f(of)e(a)g(giv)m(en)h(QP)g(\()p Fu(A;)17 b(S)6 b Fw(\).)88 b(Hence)49 b(w)m(e)f(can)g(talk)f(ab)s(out)g Fv(inde)-5 b(c)g(omp)g(osable)53 b Fw(QP-)94 2064 y(represen)m (tations.)e(Clearly)-8 b(,)34 b(the)h(righ)m(t-equiv)-5 b(alence)34 b(relation)e(resp)s(ects)37 b(direct)d(sums)h(and)g(in-)94 2180 y(decomp)s(osabilit)m(y)-8 b(.)51 b(It)36 b(is)f(also)g(immediate) e(from)i(the)h(de\014nitions)f(that)g(an)m(y)i(m)m(utation)d Fu(\026)3505 2195 y Fq(k)3583 2180 y Fw(of)94 2296 y(QP-represen)m (tations)42 b(sends)h(direct)d(sums)h(to)g(direct)f(sums.)68 b(Com)m(bining)39 b(this)i(with)f(Theo-)94 2412 y(rem)290 2412 y SDict begin H.S end 290 2412 a Fw(10.13)512 2348 y SDict begin H.R end 512 2348 a 512 2412 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.10.13) cvn H.B /ANN pdfmark end 512 2412 a Fw(,)32 b(w)m(e)i(obtain)e(the)h(follo)m(wing)c (corollary)-8 b(.)94 2432 y SDict begin H.S end 94 2432 a 94 2432 a SDict begin 14 H.A end 94 2432 a 94 2432 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.10.14) cvn H.B /DEST pdfmark end 94 2432 a 140 x FA(Corollary)39 b(10.14.)j Fv(A)n(ny)35 b(mutation)h Fu(\026)1606 2587 y Fq(k)1684 2572 y Fv(is)f(an)g(involution)g(on)g(the)g(set)h(of)f(right-e)-5 b(quivalenc)g(e)94 2689 y(classes)34 b(of)h(inde)-5 b(c)g(omp)g(osable) 32 b(de)-5 b(c)g(or)g(ate)g(d)35 b(r)-5 b(epr)g(esentations)33 b(of)i(r)-5 b(e)g(duc)g(e)g(d)35 b(QPs)f(satisfying)g Fw(\()3446 2689 y SDict begin H.S end 3446 2689 a Fw(5.1)3571 2624 y SDict begin H.R end 3571 2624 a 3571 2689 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.1) cvn H.B /ANN pdfmark end 3571 2689 a Fw(\))p Fv(.)194 2849 y Fw(W)-8 b(e)33 b(call)e(a)h(QP-represen)m(tation)h(\()p Fu(A;)17 b(S;)g(M)5 b(;)17 b(V)k Fw(\))32 b Fv(p)-5 b(ositive)40 b Fw(if)31 b Fu(V)49 b Fw(=)28 b Fr(f)p Fw(0)p Fr(g)p Fw(.)42 b(Th)m(us,)34 b(indecomp)s(os-)94 2965 y(able)k(p)s(ositiv)m(e) f(represen)m(tations)j(are)e(just)g(indecomp)s(osable)f Fr(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\)-mo)s(dules)37 b(annihilated)94 3081 y(b)m(y)g(su\016cien)m(tly)f(large)f(p)s(o)m(w)m (ers)i Fl(m)1368 3045 y Fq(n)1415 3081 y Fw(.)53 b(In)36 b(particular,)f(for)g(ev)m(ery)i(v)m(ertex)h Fu(k)s Fw(,)e(the)g Fv(simple)42 b Fw(repre-)94 3197 y(sen)m(tation)32 b Fr(S)574 3212 y Fq(k)616 3197 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\))31 b(is)g(the)g(indecomp)s(osable)f(p)s(ositiv)m(e)g(represen)m (tation)i(of)f(\()p Fu(A;)17 b(S)6 b Fw(\))30 b(suc)m(h)j(that)94 3314 y(dim)16 b Fu(M)368 3329 y Fq(i)436 3314 y Fw(=)39 b Fu(\016)594 3329 y Fq(i;k)681 3314 y Fw(.)64 b(W)-8 b(e)40 b(denote)g(b)m(y)h Fr(S)1478 3272 y Fn(\000)1471 3341 y Fq(k)1537 3314 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\))39 b(the)h(indecomp)s(osable)f(represen)m(tation)h(\()p Fu(M)5 b(;)17 b(V)k Fw(\))94 3431 y(of)39 b(\()p Fu(A;)17 b(S)6 b Fw(\))38 b(suc)m(h)j(that)e Fu(M)49 b Fw(=)38 b Fr(f)p Fw(0)p Fr(g)h Fw(and)g(dim)15 b Fu(V)1831 3446 y Fq(i)1897 3431 y Fw(=)39 b Fu(\016)2055 3446 y Fq(i;k)2141 3431 y Fw(.)63 b(W)-8 b(e)39 b(refer)h(to)e Fr(S)2829 3390 y Fn(\000)2822 3459 y Fq(k)2889 3431 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\))39 b(as)g(the)g Fv(ne)-5 b(g-)94 3547 y(ative)40 b(simple)45 b Fw(represen)m(tation)39 b(at)f Fu(k)s Fw(.)61 b(The)39 b(follo)m(wing)d(prop)s(osition)g(is)i(immediate)e(from)h(the) 94 3664 y(de\014nitions.)94 3664 y SDict begin H.S end 94 3664 a 94 3664 a SDict begin 14 H.A end 94 3664 a 94 3664 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.10.15) cvn H.B /DEST pdfmark end 94 3664 a 159 x FA(Prop)s(osition)g(10.15.)42 b Fv(A)n(ny)36 b(inde)-5 b(c)g(omp)g(osable)33 b(p)-5 b(ositive)35 b(QP-r)-5 b(epr)g(esentation)34 b(is)i(either)f(p)-5 b(osi-)94 3940 y(tive,)35 b(or)g(ne)-5 b(gative)34 b(simple.)44 b(If)34 b Fu(\026)1318 3955 y Fq(k)1360 3940 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\))28 b(=)f(\()p 1788 3860 74 4 v Fu(A;)p 1905 3860 66 4 v 17 w(S)6 b Fw(\))35 b Fv(then)f(we)h(have)94 4102 y Fw(\(10.30\))737 4102 y SDict begin H.S end 737 4102 a 737 4102 a SDict begin 14 H.A end 737 4102 a 737 4102 a SDict begin [ /View [/XYZ H.V] /Dest (equation.10.30) cvn H.B /DEST pdfmark end 737 4102 a Fu(\026)796 4117 y Fq(k)839 4102 y Fw(\()p Fr(S)937 4117 y Fq(k)980 4102 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\)\))27 b(=)g Fr(S)1474 4060 y Fn(\000)1467 4129 y Fq(k)1534 4102 y Fw(\()p 1572 4022 74 4 v Fu(A;)p 1689 4022 66 4 v 17 w(S)6 b Fw(\))p Fu(;)116 b(\026)1995 4117 y Fq(k)2037 4102 y Fw(\()p Fr(S)2142 4060 y Fn(\000)2135 4129 y Fq(k)2202 4102 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\)\))27 b(=)h Fr(S)2690 4117 y Fq(k)2733 4102 y Fw(\()p 2771 4022 74 4 v Fu(A;)p 2888 4022 66 4 v 17 w(S)5 b Fw(\);)94 4259 y Fv(and)33 b(this)f(is)h(the)g(only)f(mutation)h(that)g(inter)-5 b(changes)32 b(p)-5 b(ositive)32 b(and)g(ne)-5 b(gative)32 b(inde)-5 b(c)g(omp)g(osable)94 4375 y(r)g(epr)g(esentations.)94 4475 y SDict begin H.S end 94 4475 a 94 4475 a SDict begin 14 H.A end 94 4475 a 94 4475 a SDict begin [ /View [/XYZ H.V] /Dest (section.11) cvn H.B /DEST pdfmark end 94 4475 a 1059 4572 a Fw(11.)49 b Fx(Some)38 b(three-ver)-7 b(tex)36 b(examples)194 4747 y Fw(In)47 b(this)f(section)h(w)m(e)h(illustrate)c (the)j(action)f(of)g(m)m(utations)f(on)i(QP-represen)m(tations)g(b)m(y) 94 4863 y(some)d(examples)g(dealing)f(with)h(three-v)m(ertex)i(quiv)m (ers.)79 b(All)42 b(the)j(represen)m(tations)g(\()p Fu(M)5 b(;)17 b(V)k Fw(\))94 4979 y(considered)34 b(b)s(elo)m(w)e(will)f(b)s (e)h(p)s(ositiv)m(e,)g(i.e.,)h Fu(V)49 b Fw(=)27 b Fr(f)p Fw(0)p Fr(g)p Fw(.)94 5004 y SDict begin H.S end 94 5004 a 94 5004 a SDict begin 14 H.A end 94 5004 a 94 5004 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.11.1) cvn H.B /DEST pdfmark end 94 5004 a 135 x FA(Example)42 b(11.1.)i Fw(Let)38 b Fu(Q)f Fw(b)s(e)g(the)h(quiv)m(er)g(with)e(three)i(v)m(ertices)g(1)p Fu(;)17 b Fw(2)p Fu(;)g Fw(3)36 b(and)h(t)m(w)m(o)h(arro)m(ws)g Fu(a)d Fw(:)94 5255 y(1)28 b Fr(!)f Fw(2)33 b(and)f Fu(b)c Fw(:)g(2)g Fr(!)f Fw(3:)1854 5379 y(2)2060 5481 y Fq(b)2131 5612 y Fk(\036)p Fj(\036)2102 5582 y Fi(=)2074 5552 y(=)2045 5522 y(=)2017 5492 y(=)1989 5462 y(=)1960 5433 y(=)1932 5403 y(=)1547 5700 y Fw(1)1658 5480 y Fq(a)1825 5403 y Fk(@)p Fj(@)1796 5433 y Fi(\001)1768 5463 y(\001)1739 5493 y(\001)1711 5523 y(\001)1683 5552 y(\001)1654 5582 y(\001)1626 5612 y(\001)2160 5700 y Fw(3)p eop end end %%Page: 50 50 TeXDict begin HPSdict begin 50 49 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.50) cvn H.B /DEST pdfmark end 94 218 a Fs(50)492 b(HARM)25 b(DERKSEN,)f(JERZY)i(WEYMAN,)f(AND)f(ANDREI)g(ZELEVINSKY)94 422 y Fw(Since)34 b Fu(Q)g Fw(is)f(acyclic,)h(the)g(only)f(QP)h(on)g (it)f(is)g(\()p Fu(A;)17 b Fw(0\).)46 b(W)-8 b(e)34 b(ha)m(v)m(e)h Fu(\026)2570 437 y Fp(2)2609 422 y Fw(\()p Fu(A;)17 b Fw(0\))29 b(=)h(\()p 3024 342 74 4 v Fu(A;)p 3141 342 66 4 v 17 w(S)5 b Fw(\),)34 b(where)p 3588 342 74 4 v 35 w Fu(A)94 543 y Fw(is)f(the)g(arro)m(w)f(span)h(of)g(the)g(quiv)m (er)p 1437 463 78 4 v 33 w Fu(Q)g Fw(giv)m(en)f(b)m(y)1854 713 y(2)1622 816 y Fq(a)1659 792 y Fg(?)1625 947 y Fk(\000)p Fj(\000)p Fi(\001)1654 917 y(\001)1682 887 y(\001)1710 857 y(\001)1739 828 y(\001)1767 798 y(\001)1796 768 y(\001)1547 1035 y Fw(1)1825 1082 y Fp([)p Fq(ba)p Fp(])2131 1003 y Fk(/)p Fj(/)p 1625 1005 506 4 v 2160 1035 a Fw(3)2060 816 y Fq(b)2090 792 y Fg(?)1931 738 y Fk(^)p Fj(^)p Fi(=)1960 767 y(=)1988 797 y(=)2017 827 y(=)2045 857 y(=)2073 887 y(=)2102 917 y(=)94 1271 y Fw(and)p 285 1191 66 4 v 34 w Fu(S)i Fw(=)29 b Fu(b)525 1235 y Fq(?)565 1271 y Fw([)p Fu(ba)p Fw(])p Fu(a)762 1235 y Fq(?)803 1271 y Fw(.)45 b(Th)m(us,)35 b(p)s(ositiv)m(e)e(represen)m(tations)h(of)f(\()p Fu(A;)17 b Fw(0\))32 b(are)i(the)f(represen)m(tations)i(of)94 1392 y(the)g(quiv)m(er)g Fu(Q)p Fw(,)g(while)e(p)s(ositiv)m(e)h (represen)m(tations)h(of)f(\()p 2147 1312 74 4 v Fu(A;)p 2264 1312 66 4 v 17 w(S)6 b Fw(\))34 b(are)g(the)h(represen)m(tations)g (of)f(the)94 1513 y(quiv)m(er)p 390 1433 78 4 v 34 w Fu(Q)f Fw(satisfying)e(the)i(relations)94 1676 y(\(11.1\))1325 1676 y SDict begin H.S end 1325 1676 a 1325 1676 a SDict begin 14 H.A end 1325 1676 a 1325 1676 a SDict begin [ /View [/XYZ H.V] /Dest (equation.11.1) cvn H.B /DEST pdfmark end 1325 1676 a Fu(b)1366 1635 y Fq(?)1406 1676 y Fw([)p Fu(ba)p Fw(])28 b(=)g([)p Fu(ba)p Fw(])p Fu(a)1881 1635 y Fq(?)1949 1676 y Fw(=)g Fu(a)2104 1635 y Fq(?)2143 1676 y Fu(b)2184 1635 y Fq(?)2252 1676 y Fw(=)f(0)p Fu(:)94 1839 y Fw(In)48 b(view)f(of)f(Corollary)1041 1839 y SDict begin H.S end 1041 1839 a Fw(10.14)1263 1775 y SDict begin H.R end 1263 1775 a 1263 1839 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.10.14) cvn H.B /ANN pdfmark end 1263 1839 a 47 w Fw(and)h(Prop)s(osition)2053 1839 y SDict begin H.S end 2053 1839 a Fw(10.15)2275 1775 y SDict begin H.R end 2275 1775 a 2275 1839 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.10.15) cvn H.B /ANN pdfmark end 2275 1839 a Fw(,)k(the)c(m)m(utation)e Fu(\026)3028 1854 y Fp(2)3114 1839 y Fw(establishes)j(a)94 1956 y(bijection)28 b(b)s(et)m(w)m(een)i(the)f(set)g(of)e(righ)m(t-equiv)-5 b(alence)28 b(classes)h(of)f(indecomp)s(osable)f(p)s(ositiv)m(e)g(rep-) 94 2072 y(resen)m(tations)e(of)e(\()p Fu(A;)17 b Fw(0\))23 b(di\013eren)m(t)h(from)f(the)h(simple)e(represen)m(tation)j Fr(S)2734 2087 y Fp(2)2773 2072 y Fw(,)h(and)e(the)g(same)f(set)i(for) 94 2193 y(\()p 132 2113 74 4 v Fu(A)q(;)p 250 2113 66 4 v 17 w(S)5 b Fw(\).)42 b(Since)27 b Fu(Q)g Fw(is)g(a)g(Dynkin)g(quiv) m(er)h(of)f(t)m(yp)s(e)h Fu(A)1967 2208 y Fp(3)2007 2193 y Fw(,)g(b)m(y)g(Gabriel's)e(theorem,)i(an)f(indecomp)s(os-)94 2309 y(able)d(p)s(ositiv)m(e)g(represen)m(tation)i Fu(M)35 b Fw(of)24 b(\()p Fu(A;)17 b Fw(0\))24 b(is)g(uniquely)h(up)g(to)f(an)g (isomorphism)f(determined)94 2426 y(b)m(y)41 b(its)e(dimension)e(v)m (ector)k FA(dim)15 b Fu(M)50 b Fw(=)39 b(\(dim)15 b Fu(M)1920 2441 y Fp(1)1960 2426 y Fu(;)i Fw(dim)e Fu(M)2277 2441 y Fp(2)2317 2426 y Fu(;)i Fw(dim)e Fu(M)2634 2441 y Fp(3)2674 2426 y Fw(\),)41 b(and)e(these)i(dimension)94 2542 y(v)m(ectors)30 b(are)d(the)h(p)s(ositiv)m(e)f(ro)s(ots)h(of)f(t)m(yp)s(e)h Fu(A)1733 2557 y Fp(3)1800 2542 y Fw(\(note)g(that)g(in)e(this)i(case,) h(the)f(righ)m(t-equiv)-5 b(alence)94 2658 y(classes)40 b(are)e(the)g(same)g(as)h(isomorphism)d(classes\).)61 b(Computing)37 b(the)h(images)f(of)h(these)h(rep-)94 2774 y(resen)m(tations)33 b(under)g Fu(\026)969 2789 y Fp(2)1008 2774 y Fw(,)f(w)m(e)h(obtain)d(the)i(corresp)s(ondence)i(b) s(et)m(w)m(een)g(the)e(dimension)f(v)m(ectors)94 2890 y(giv)m(en)38 b(in)e(T)-8 b(able)745 2890 y SDict begin H.S end 745 2890 a Fw(1)794 2826 y SDict begin H.R end 794 2826 a 794 2890 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (table.1) cvn H.B /ANN pdfmark end 794 2890 a Fw(.)114 b(W)-8 b(e)37 b(conclude)h(that)f(an)g(indecomp)s(osable)f(p)s(ositiv)m (e)g(represen)m(tation)i(of)p 697 3041 2363 4 v 695 3157 4 117 v 1260 3157 V 1620 3157 V 1979 3157 V 2339 3157 V 2698 3157 V 3057 3157 V 695 3213 V 825 3178 a FA(dim)16 b Fu(M)p 1260 3213 V 181 w Fw(\(1,0,0\))p 1620 3213 V 82 w(\(0,0,1\))p 1979 3213 V 82 w(\(1,1,0\))p 2339 3213 V 83 w(\(0,1,1\))p 2698 3213 V 82 w(\(1,1,1\))p 3057 3213 V 697 3217 2363 4 v 695 3333 4 117 v 1260 3333 V 1620 3333 V 1979 3333 V 2339 3333 V 2698 3333 V 3057 3333 V 695 3419 4 147 v 738 3354 a FA(dim)g Fu(\026)1001 3369 y Fp(2)1040 3354 y Fw(\()p Fu(M)10 b Fw(\))p 1260 3419 V 84 w(\(1,1,0\))p 1620 3419 V 82 w(\(0,1,1\))p 1979 3419 V 82 w(\(1,0,0\))p 2339 3419 V 83 w(\(0,0,1\))p 2698 3419 V 82 w(\(1,0,1\))p 3057 3419 V 697 3422 2363 4 v 314 3561 a Fx(T)-9 b(able)38 b(1.)783 3561 y SDict begin H.S end 783 3561 a 783 3561 a SDict begin H.R end 783 3561 a 783 3561 a SDict begin [ /View [/XYZ H.V] /Dest (table.1) cvn H.B /DEST pdfmark end 783 3561 a Fw(Indecomp)s (osable)32 b(represen)m(tations)i(for)e Fu(A)2391 3576 y Fp(3)2463 3561 y Fw(and)h(the)g(cyclic)f(triangle.)94 3824 y(\()p 132 3744 74 4 v Fu(A)q(;)p 250 3744 66 4 v 17 w(S)5 b Fw(\))37 b(is)g(uniquely)g(up)h(to)f(righ)m(t-equiv)-5 b(alence)36 b(determined)h(b)m(y)h(its)f(dimension)f(v)m(ector,)j(and) 94 3940 y(these)e(dimension)e(v)m(ectors)i(are)f(giv)m(en)g(in)f(the)h (second)h(line)d(of)h(T)-8 b(able)2749 3940 y SDict begin H.S end 2749 3940 a Fw(1)2798 3876 y SDict begin H.R end 2798 3876 a 2798 3940 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (table.1) cvn H.B /ANN pdfmark end 2798 3940 a Fw(,)36 b(with)g(the)g(exception)94 4056 y(of)d FA(dim)15 b Fr(S)469 4071 y Fp(2)536 4056 y Fw(=)28 b(\(0)p Fu(;)17 b Fw(1)p Fu(;)g Fw(0\).)94 4143 y SDict begin H.S end 94 4143 a 94 4143 a SDict begin 14 H.A end 94 4143 a 94 4143 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.11.2) cvn H.B /DEST pdfmark end 94 4143 a 91 x FA(Example)33 b(11.2.)40 b Fw(No)m(w)29 b(let)g Fu(Q)h Fw(b)s(e)f(the)h(quiv)m(er)g(with)f (three)h(v)m(ertices)g(1)p Fu(;)17 b Fw(2)p Fu(;)g Fw(3)28 b(and)i(three)g(arro)m(ws)94 4350 y Fu(a)e Fw(:)g(1)g Fr(!)f Fw(2,)32 b Fu(b)c Fw(:)g(2)g Fr(!)f Fw(3,)32 b(and)h Fu(c)28 b Fw(:)f(1)h Fr(!)f Fw(3:)1854 4521 y(2)2060 4623 y Fq(b)2131 4754 y Fk(\036)p Fj(\036)2102 4724 y Fi(=)2074 4694 y(=)2045 4664 y(=)2017 4634 y(=)1989 4605 y(=)1960 4575 y(=)1932 4545 y(=)1547 4842 y Fw(1)1658 4622 y Fq(a)1825 4545 y Fk(@)p Fj(@)1796 4575 y Fi(\001)1768 4605 y(\001)1739 4635 y(\001)1711 4665 y(\001)1683 4694 y(\001)1654 4724 y(\001)1626 4754 y(\001)1863 4868 y Fq(c)2131 4810 y Fk(/)p Fj(/)p 1625 4812 506 4 v 2160 4842 a Fw(3)94 5040 y(Again,)j(the)h(only)e(QP)i(on)f Fu(Q)g Fw(is)g(\()p Fu(A;)17 b Fw(0\).)42 b(W)-8 b(e)31 b(ha)m(v)m(e)g Fu(\026)2048 5055 y Fp(2)2087 5040 y Fw(\()p Fu(A;)17 b Fw(0\))28 b(=)f(\()p 2498 4960 74 4 v Fu(A;)p 2615 4960 66 4 v 17 w(S)6 b Fw(\),)30 b(where)p 3056 4960 74 4 v 32 w Fu(A)g Fw(is)g(the)g(arro)m(w)94 5161 y(span)k(of)e(the)h(quiv)m(er)p 897 5081 78 4 v 33 w Fu(Q)g Fw(giv)m(en)g(b)m(y)1854 5332 y(2)1622 5434 y Fq(a)1659 5410 y Fg(?)1625 5565 y Fk(\000)p Fj(\000)p Fi(\001)1654 5535 y(\001)1682 5505 y(\001)1710 5476 y(\001)1739 5446 y(\001)1767 5416 y(\001)1796 5386 y(\001)1547 5653 y Fw(1)1863 5700 y Fq(c)2131 5643 y Fk(/)p Fj(/)p 1625 5644 506 4 v 1825 5554 a Fp([)p Fq(ba)p Fp(])2131 5600 y Fk(/)p Fj(/)p 1625 5601 V 2160 5653 a Fw(3)2060 5434 y Fq(b)2090 5410 y Fg(?)1931 5356 y Fk(^)p Fj(^)p Fi(=)1960 5386 y(=)1988 5415 y(=)2017 5445 y(=)2045 5475 y(=)2073 5505 y(=)2102 5535 y(=)p eop end end %%Page: 51 51 TeXDict begin HPSdict begin 51 50 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.51) cvn H.B /DEST pdfmark end 94 218 a 1179 w Fs(QUIVERS)31 b(WITH)i(POTENTIALS)g(I)1102 b(51)94 422 y Fw(and)p 281 342 66 4 v 30 w Fu(S)34 b Fw(=)27 b Fu(b)519 386 y Fq(?)559 422 y Fw([)p Fu(ba)p Fw(])p Fu(a)756 386 y Fq(?)797 422 y Fw(.)42 b(Again,)29 b(p)s(ositiv)m(e)g (represen)m(tations)i(of)e(\()p Fu(A;)17 b Fw(0\))29 b(are)h(the)g(represen)m(tations)h(of)94 543 y(the)k(quiv)m(er)g Fu(Q)p Fw(,)g(while)e(p)s(ositiv)m(e)h(represen)m(tations)h(of)f(\()p 2147 463 74 4 v Fu(A;)p 2264 463 66 4 v 17 w(S)6 b Fw(\))34 b(are)g(the)h(represen)m(tations)g(of)f(the)94 664 y(quiv)m(er)p 390 584 78 4 v 34 w Fu(Q)f Fw(satisfying)e(the)i(relations)e(\()1536 664 y SDict begin H.S end 1536 664 a Fw(11.1)1710 600 y SDict begin H.R end 1710 600 a 1710 664 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.11.1) cvn H.B /ANN pdfmark end 1710 664 a Fw(\).)194 781 y(W)-8 b(e)30 b(consider)h(indecomp)s (osable)d(p)s(ositiv)m(e)i(represen)m(tations)h(of)e(\()p Fu(A;)17 b Fw(0\))30 b(with)f(the)i(dimension)94 897 y(v)m(ector)38 b(\()p Fu(n;)17 b(n;)g(n)p Fw(\))37 b(for)f(some)h Fu(n)e Fr(\025)g Fw(1.)55 b(Assume)38 b(that)e Fu(K)44 b Fw(is)36 b(algebraically)e(closed.)56 b(Since)37 b Fu(Q)g Fw(is)94 1031 y(an)30 b(extended)i(Dynkin)e(quiv)m(er)h(of)f(t)m (yp)s(e)h Fu(A)1674 980 y Fp(\(1\))1674 1055 y(2)1768 1031 y Fw(,)g(and)f(\()p Fu(n;)17 b(n;)g(n)p Fw(\))30 b(is)f(an)h(isotropic)f(imaginary)e(ro)s(ot,)94 1147 y(b)m(y)k(Kac's)g(extension)f(of)g(Gabriel's)e(theorem,)j(the)f (isomorphism)d(classes)k(of)f(indecomp)s(osable)94 1263 y Fu(Q)p Fw(-represen)m(tations)c(of)f(this)f(dimension)g(form)f(a)i (1-parametric)d(family)-8 b(.)39 b(An)25 b(easy)h(c)m(hec)m(k)h(sho)m (ws)94 1379 y(that)i(these)g(represen)m(tations)h(break)f(in)m(to)e (three)i(righ)m(t-equiv)-5 b(alence)28 b(classes.)43 b(Their)28 b(represen-)94 1496 y(tativ)m(es)h(can)f(b)s(e)g(describ)s (ed)h(as)f(follo)m(ws.)41 b(F)-8 b(or)27 b(eac)m(h)i(of)f(them)g(w)m(e) h(ha)m(v)m(e)g Fu(M)2796 1511 y Fp(1)2864 1496 y Fw(=)e Fu(M)3061 1511 y Fp(2)3129 1496 y Fw(=)g Fu(M)3326 1511 y Fp(3)3394 1496 y Fw(=)g Fu(K)3587 1460 y Fq(n)3634 1496 y Fw(,)94 1612 y(and)k(t)m(w)m(o)f(of)g(the)g(maps)g Fu(a)1041 1627 y Fq(M)1120 1612 y Fu(;)17 b(b)1205 1627 y Fq(M)1284 1612 y Fu(;)g(c)1370 1627 y Fq(M)1479 1612 y Fw(are)30 b(equal)g(to)f(the)i(iden)m(tit)m(y)f(map)f Fu(I)8 b Fw(,)30 b(while)f(the)i(third)e(one)94 1728 y(is)41 b(the)h(nilp)s(oten)m(t)e(Jordan)i(blo)s(c)m(k)f Fu(N)10 b Fw(.)71 b(If)41 b Fu(a)1749 1743 y Fq(M)1871 1728 y Fw(=)h Fu(N)52 b Fw(\(resp.)71 b Fu(b)2469 1743 y Fq(M)2591 1728 y Fw(=)43 b Fu(N)10 b Fw(,)44 b Fu(c)2911 1743 y Fq(M)3033 1728 y Fw(=)e Fu(N)10 b Fw(\))42 b(then)g(w)m(e)94 1844 y(denote)33 b(the)f(corresp)s(onding)f Fu(Q)p Fw(-represen)m (tation)h(b)m(y)h Fu(M)10 b Fw(\()p Fu(a)p Fw(\))32 b(\(resp.)44 b Fu(M)10 b Fw(\()p Fu(b)p Fw(\),)33 b Fu(M)10 b Fw(\()p Fu(c)p Fw(\)\).)44 b(In)32 b(view)g(of)94 1966 y(\()132 1966 y SDict begin H.S end 132 1966 a Fw(10.7)306 1901 y SDict begin H.R end 306 1901 a 306 1966 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.7) cvn H.B /ANN pdfmark end 306 1966 a Fw(\),)d(if)d Fu(M)39 b Fw(is)27 b(one)i(of)e(these)i (represen)m(tations)g(then)g Fu(\026)2184 1981 y Fp(2)2223 1966 y Fw(\()p Fu(M)10 b Fw(\))28 b(=)p 2535 1886 105 4 v 28 w Fu(M)38 b Fw(is)28 b(p)s(ositiv)m(e,)g(and)g(w)m(e)h(ha)m(v)m (e)p 1353 2048 V 1353 2128 a Fu(M)1458 2143 y Fp(2)1525 2128 y Fw(=)f(cok)m(er)33 b Fu(b)1924 2143 y Fq(M)2026 2128 y Fr(\010)23 b Fw(k)m(er)17 b Fu(a)2323 2143 y Fq(M)94 2287 y Fw(\(note)36 b(that)f(since)g Fu(S)j Fw(=)32 b(0,)j(w)m(e)h(ha)m (v)m(e)h Fu(\015)g Fw(=)32 b(0\).)51 b(It)35 b(follo)m(ws)e(that)p 2503 2200 223 4 v 35 w Fu(M)10 b Fw(\()p Fu(c)p Fw(\))36 b(has)g(dimension)d(v)m(ector)94 2403 y(\()p Fu(n;)17 b Fw(0)p Fu(;)g(n)p Fw(\),)32 b(with)g(the)g(maps)f([)p Fu(ba)p Fw(])p 1271 2375 165 3 v 31 x Fq(M)7 b Fp(\()p Fq(c)p Fp(\))1468 2403 y Fu(;)17 b(c)p 1554 2375 V 31 x Fq(M)7 b Fp(\()p Fq(c)p Fp(\))1746 2403 y Fw(:)28 b Fu(K)1891 2367 y Fq(n)1966 2403 y Fr(!)f Fu(K)2183 2367 y Fq(N)2282 2403 y Fw(giv)m(en)32 b(b)m(y)h([)p Fu(ba)p Fw(])p 2817 2375 V 31 x Fq(M)7 b Fp(\()p Fq(c)p Fp(\))3010 2403 y Fw(=)28 b Fu(I)8 b(;)17 b(c)p 3251 2375 V 31 x Fq(M)7 b Fp(\()p Fq(c)p Fp(\))3442 2403 y Fw(=)28 b Fu(N)10 b Fw(.)94 2553 y(Also)36 b(b)s(oth)g(represen)m(tations)p 1231 2467 232 4 v 37 w Fu(M)10 b Fw(\()p Fu(a)p Fw(\))37 b(and)p 1692 2467 223 4 v 36 w Fu(M)10 b Fw(\()p Fu(b)p Fw(\))37 b(ha)m(v)m(e)h(dimension)c(v)m(ector)k(\()p Fu(n;)17 b Fw(1)p Fu(;)g(n)p Fw(\).)53 b(In)36 b(eac)m(h)94 2670 y(of)d(them,)g(the)g(arro)m(ws)h([)p Fu(ba)p Fw(])g(and)f Fu(c)g Fw(act)g(as)g([)p Fu(ba)p Fw(])c(=)f Fu(N)5 b(;)17 b(c)28 b Fw(=)h Fu(I)8 b Fw(.)44 b(W)-8 b(e)34 b(also)e(ha)m(v)m(e)i Fu(b)3026 2633 y Fq(?)p 3026 2655 172 3 v 3026 2714 a(M)7 b Fp(\()p Fq(a)p Fp(\))3226 2670 y Fw(=)28 b(0,)33 b(while)94 2808 y(the)h(map)f Fu(a)532 2772 y Fq(?)p 532 2794 V 532 2853 a(M)7 b Fp(\()p Fq(a)p Fp(\))732 2808 y Fw(:)30 b Fu(K)36 b Fr(!)28 b Fu(K)1126 2772 y Fq(n)1207 2808 y Fw(has)33 b(im)f Fu(a)1574 2772 y Fq(?)p 1574 2794 V 1574 2853 a(M)7 b Fp(\()p Fq(a)p Fp(\))1775 2808 y Fw(=)29 b(k)m(er)17 b Fu(N)10 b Fw(;)35 b(similarly)-8 b(,)29 b Fu(a)2645 2772 y Fq(?)p 2645 2794 165 3 v 2645 2853 a(M)7 b Fp(\()p Fq(b)p Fp(\))2839 2808 y Fw(=)29 b(0,)k(while)f(the)i(map)94 2947 y Fu(b)135 2911 y Fq(?)p 135 2933 V 135 2991 a(M)7 b Fp(\()p Fq(b)p Fp(\))328 2947 y Fw(:)27 b Fu(K)472 2911 y Fq(n)547 2947 y Fr(!)h Fu(K)39 b Fw(has)33 b(k)m(er)18 b Fu(b)1159 2911 y Fq(?)p 1159 2933 172 3 v 1159 2991 a(M)7 b Fp(\()p Fq(a)p Fp(\))1358 2947 y Fw(=)28 b(im)j Fu(N)10 b Fw(.)94 3066 y SDict begin H.S end 94 3066 a 94 3066 a SDict begin 14 H.A end 94 3066 a 94 3066 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.11.3) cvn H.B /DEST pdfmark end 94 3066 a 83 x FA(Example)54 b(11.3.)c Fw(Our)e(last)f(example)g(deals)h(with)f(the)i(QP)f(\()p Fu(A;)17 b(S)6 b Fw(\))47 b(from)f(Example)3510 3149 y SDict begin H.S end 3510 3149 a Fw(8.6)3634 3085 y SDict begin H.R end 3634 3085 a 3634 3149 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.8.6) cvn H.B /ANN pdfmark end 3634 3149 a Fw(.)94 3266 y(Th)m(us,)38 b(the)d(quiv)m(er)h(in)e (question)i(has)f(three)h(v)m(ertices)g(1)p Fu(;)17 b Fw(2)p Fu(;)g Fw(3)34 b(and)h(six)g(arro)m(ws)h Fu(a)3109 3281 y Fp(1)3148 3266 y Fu(;)17 b(a)3243 3281 y Fp(2)3315 3266 y Fw(:)31 b(1)h Fr(!)g Fw(2,)94 3382 y Fu(b)135 3397 y Fp(1)175 3382 y Fu(;)17 b(b)260 3397 y Fp(2)328 3382 y Fw(:)27 b(2)h Fr(!)f Fw(3)32 b(and)h Fu(c)899 3397 y Fp(1)939 3382 y Fu(;)17 b(c)1025 3397 y Fp(2)1091 3382 y Fw(:)28 b(3)g Fr(!)f Fw(1;)32 b(and)h(the)g(p)s(oten)m(tial)e Fu(S)38 b Fw(is)32 b(giv)m(en)h(b)m(y)g(\()2852 3382 y SDict begin H.S end 2852 3382 a Fw(8.1)2977 3318 y SDict begin H.R end 2977 3318 a 2977 3382 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.8.1) cvn H.B /ANN pdfmark end 2977 3382 a Fw(\).)1854 3549 y(2)2076 3628 y Fq(b)2106 3637 y Fo(1)2146 3768 y Fk(\036)p Fj(\036)2118 3738 y Fi(=)2089 3708 y(=)2061 3678 y(=)2033 3648 y(=)2004 3619 y(=)1976 3589 y(=)1947 3559 y(=)1922 3765 y Fq(b)1952 3774 y Fo(2)2115 3798 y Fk(\036)p Fj(\036)2087 3767 y Fi(=)2058 3737 y(=)2030 3708 y(=)2001 3678 y(=)1973 3648 y(=)1945 3618 y(=)1916 3588 y(=)1547 3871 y Fw(1)1608 3627 y Fq(a)1645 3636 y Fo(1)1809 3559 y Fk(@)p Fj(@)1780 3589 y Fi(\001)1752 3619 y(\001)1724 3649 y(\001)1695 3679 y(\001)1667 3708 y(\001)1638 3738 y(\001)1610 3768 y(\001)1770 3748 y Fq(a)1807 3757 y Fo(2)1840 3588 y Fk(@)p Fj(@)1812 3619 y Fi(\001)1783 3648 y(\001)1755 3678 y(\001)1727 3708 y(\001)1698 3738 y(\001)1670 3768 y(\001)1641 3797 y(\001)2160 3871 y Fw(3)1845 3779 y Fq(c)1876 3788 y Fo(2)1625 3818 y Fk(o)p Fj(o)p 1625 3819 506 4 v 1845 3918 a Fq(c)1876 3927 y Fo(1)1625 3860 y Fk(o)p Fj(o)p 1625 3862 V 94 4081 a Fw(T)-8 b(o)24 b(sp)s(ecify)g(a)f(p)s(ositiv)m(e)g(represen)m(tation)h Fu(M)34 b Fw(of)23 b(\()p Fu(A;)17 b(S)6 b Fw(\),)25 b(w)m(e)f(need)g(to)g(de\014ne)g(three)g(v)m(ector)h(spaces)94 4197 y Fu(M)188 4212 y Fp(1)228 4197 y Fu(;)17 b(M)366 4212 y Fp(2)406 4197 y Fu(;)g(M)544 4212 y Fp(3)583 4197 y Fw(,)33 b(and)g(six)g(linear)e(maps)i(\()p Fu(a)1599 4212 y Fp(1)1639 4197 y Fw(\))1677 4212 y Fq(M)1756 4197 y Fu(;)17 b Fw(\()p Fu(a)1889 4212 y Fp(2)1928 4197 y Fw(\))1966 4212 y Fq(M)2073 4197 y Fw(:)28 b Fu(M)2222 4212 y Fp(1)2290 4197 y Fr(!)g Fu(M)2512 4212 y Fp(2)2552 4197 y Fw(,)33 b(\()p Fu(b)2691 4212 y Fp(1)2731 4197 y Fw(\))2769 4212 y Fq(M)2848 4197 y Fu(;)17 b Fw(\()p Fu(b)2971 4212 y Fp(2)3010 4197 y Fw(\))3048 4212 y Fq(M)3155 4197 y Fw(:)29 b Fu(M)3305 4212 y Fp(2)3373 4197 y Fr(!)f Fu(M)3595 4212 y Fp(3)3634 4197 y Fw(,)94 4313 y(and)33 b(\()p Fu(c)364 4328 y Fp(1)403 4313 y Fw(\))441 4328 y Fq(M)520 4313 y Fu(;)17 b Fw(\()p Fu(c)644 4328 y Fp(2)683 4313 y Fw(\))721 4328 y Fq(M)828 4313 y Fw(:)28 b Fu(M)977 4328 y Fp(3)1044 4313 y Fr(!)g Fu(M)1266 4328 y Fp(1)1305 4313 y Fw(.)44 b(In)33 b(our)f(case,)h Fu(J)9 b Fw(\()p Fu(S)d Fw(\))32 b(is)g(the)h(closure)g(of)f(the)h(ideal)e(in)g Fu(R)q Fr(hh)p Fu(A)p Fr(ii)94 4429 y Fw(generated)j(b)m(y)f(six)g (elemen)m(ts)1249 4587 y Fu(c)1291 4602 y Fp(1)1331 4587 y Fu(b)1372 4602 y Fp(1)1412 4587 y Fu(;)17 b(b)1497 4602 y Fp(1)1536 4587 y Fu(a)1587 4602 y Fp(1)1627 4587 y Fu(;)g(a)1722 4602 y Fp(1)1761 4587 y Fu(c)1803 4602 y Fp(1)1843 4587 y Fu(;)g(c)1929 4602 y Fp(2)1968 4587 y Fu(b)2009 4602 y Fp(2)2049 4587 y Fu(;)g(b)2134 4602 y Fp(2)2173 4587 y Fu(a)2224 4602 y Fp(2)2264 4587 y Fu(;)g(a)2359 4602 y Fp(2)2398 4587 y Fu(c)2440 4602 y Fp(2)2480 4587 y Fu(:)94 4744 y Fw(Th)m(us,)35 b(all)30 b(the)j(comp)s(ositions)e(\()p Fu(c)1336 4759 y Fp(1)1375 4744 y Fw(\))1413 4759 y Fq(M)1492 4744 y Fw(\()p Fu(b)1571 4759 y Fp(1)1611 4744 y Fw(\))1649 4759 y Fq(M)1728 4744 y Fu(;)17 b(:)g(:)g(:)f(;)h Fw(\()p Fu(a)2036 4759 y Fp(2)2075 4744 y Fw(\))2113 4759 y Fq(M)2192 4744 y Fw(\()p Fu(c)2272 4759 y Fp(2)2311 4744 y Fw(\))2349 4759 y Fq(M)2461 4744 y Fw(m)m(ust)32 b(b)s(e)h(equal)g(to)f(0.)194 4860 y(W)-8 b(e)42 b(\014rst)g(consider)f(the)h(indecomp)s(osable)e(p)s (ositiv)m(e)h(represen)m(tation)h Fu(M)52 b Fw(of)41 b(\()p Fu(A;)17 b(S)6 b Fw(\))41 b(giv)m(en)94 4977 y(b)m(y:)94 5134 y(\(11.2\))858 5134 y SDict begin H.S end 858 5134 a 858 5134 a SDict begin 14 H.A end 858 5134 a 858 5134 a SDict begin [ /View [/XYZ H.V] /Dest (equation.11.2) cvn H.B /DEST pdfmark end 858 5134 a Fu(M)952 5149 y Fp(1)1019 5134 y Fw(=)28 b Fu(M)1217 5149 y Fp(2)1284 5134 y Fw(=)g Fu(K)r(;)114 b(M)1708 5149 y Fp(3)1775 5134 y Fw(=)28 b(0;)114 b(\()p Fu(a)2158 5149 y Fp(1)2197 5134 y Fw(\))2235 5149 y Fq(M)2342 5134 y Fw(=)28 b(\()p Fu(a)2535 5149 y Fp(2)2574 5134 y Fw(\))2612 5149 y Fq(M)2719 5134 y Fw(=)f(1)p Fu(:)94 5296 y Fw(Let)49 b(us)g(compute)f Fu(\026)896 5311 y Fp(2)935 5296 y Fw(\()p Fu(M)10 b Fw(\))56 b(=)e(\()p 1339 5216 105 4 v Fu(M)11 b(;)p 1488 5216 79 4 v 17 w(V)21 b Fw(\).)90 b(First)48 b(of)g(all,)i(the)f(QP)f Fu(\026)2717 5311 y Fp(2)2756 5296 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\))54 b(=)h(\()p 3238 5216 74 4 v Fu(A;)p 3355 5216 66 4 v 17 w(S)5 b Fw(\))48 b(w)m(as)94 5418 y(computed)36 b(in)e(Example)1070 5418 y SDict begin H.S end 1070 5418 a Fw(8.6)1195 5353 y SDict begin H.R end 1195 5353 a 1195 5418 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.8.6) cvn H.B /ANN pdfmark end 1195 5418 a Fw(:)48 b(recall)34 b(that)h(the)g(arro)m(ws)h(in)p 2348 5338 74 4 v 34 w Fu(A)f Fw(are)g Fu(a)2672 5381 y Fq(?)2672 5442 y Fp(1)2712 5418 y Fu(;)17 b(a)2807 5381 y Fq(?)2807 5442 y Fp(2)2846 5418 y Fu(;)g(b)2931 5381 y Fq(?)2931 5442 y Fp(1)2971 5418 y Fu(;)g(b)3056 5381 y Fq(?)3056 5442 y Fp(2)3095 5418 y Fu(;)g Fw([)p Fu(b)3207 5433 y Fp(1)3247 5418 y Fu(a)3298 5433 y Fp(2)3338 5418 y Fw(])p Fu(;)g Fw([)p Fu(b)3477 5433 y Fp(2)3517 5418 y Fu(a)3568 5433 y Fp(1)3607 5418 y Fw(],)94 5539 y(and)33 b(the)g(p)s(oten)m(tial)p 864 5459 66 4 v 31 w Fu(S)38 b Fw(is)32 b(giv)m(en)h(b)m(y)p 1309 5616 V 1309 5696 a Fu(S)g Fw(=)28 b([)p Fu(b)1574 5711 y Fp(1)1614 5696 y Fu(a)1665 5711 y Fp(2)1704 5696 y Fw(])p Fu(a)1782 5655 y Fq(?)1782 5721 y Fp(2)1822 5696 y Fu(b)1863 5655 y Fq(?)1863 5721 y Fp(1)1925 5696 y Fw(+)22 b([)p Fu(b)2091 5711 y Fp(2)2131 5696 y Fu(a)2182 5711 y Fp(1)2222 5696 y Fw(])p Fu(a)2300 5655 y Fq(?)2300 5721 y Fp(1)2339 5696 y Fu(b)2380 5655 y Fq(?)2380 5721 y Fp(2)2420 5696 y Fu(:)p eop end end %%Page: 52 52 TeXDict begin HPSdict begin 52 51 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.52) cvn H.B /DEST pdfmark end 94 218 a Fs(52)492 b(HARM)25 b(DERKSEN,)f(JERZY)i(WEYMAN,)f(AND)f(ANDREI)g(ZELEVINSKY)94 422 y Fw(T)-8 b(o)33 b(compute)p 634 342 105 4 v 33 w Fu(M)43 b Fw(and)p 961 342 79 4 v 33 w Fu(V)21 b Fw(,)33 b(w)m(e)g(apply)g(\()1549 422 y SDict begin H.S end 1549 422 a Fw(10.6)1722 358 y SDict begin H.R end 1722 358 a 1722 422 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.6) cvn H.B /ANN pdfmark end 1722 422 a Fw(\))g(and)f(\()2020 422 y SDict begin H.S end 2020 422 a Fw(10.7)2194 358 y SDict begin H.R end 2194 358 a 2194 422 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.7) cvn H.B /ANN pdfmark end 2194 422 a Fw(\))g(to)g(the)h(triangle)e(\()2947 422 y SDict begin H.S end 2947 422 a Fw(10.5)3120 358 y SDict begin H.R end 3120 358 a 3120 422 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.5) cvn H.B /ANN pdfmark end 3120 422 a Fw(\))i(giv)m(en)f(b)m(y)772 581 y Fu(M)866 596 y Fp(in)957 581 y Fw(=)27 b Fu(K)1150 540 y Fp(2)1190 581 y Fu(;)50 b(M)1361 596 y Fq(k)1431 581 y Fw(=)28 b Fu(M)1629 596 y Fp(2)1696 581 y Fw(=)g Fu(K)r(;)50 b(M)2056 596 y Fp(out)2190 581 y Fw(=)27 b Fr(f)p Fw(0)p Fr(g)p Fu(;)50 b(\013)28 b Fw(=)2712 501 y Fm(\000)2758 579 y Fw(1)83 b(1)2939 501 y Fm(\001)94 754 y Fw(\(so)36 b(w)m(e)h(ha)m(v)m(e)g Fu(\014)i Fw(=)33 b(0)i(and)h Fu(\015)i Fw(=)33 b(0\).)53 b(It)35 b(follo)m(ws)g(that)p 2121 674 79 4 v 35 w Fu(V)55 b Fw(=)33 b Fr(f)p Fw(0)p Fr(g)p Fw(,)j(i.e.,)g Fu(\026)2801 769 y Fp(2)2840 754 y Fw(\()p Fu(M)10 b Fw(\))36 b(is)g(p)s(ositiv)m(e;)g(w)m(e)94 875 y(also)c(ha)m(v)m(e)p 515 795 105 4 v 34 w Fu(M)620 890 y Fp(1)687 875 y Fw(=)27 b Fu(M)884 890 y Fp(1)952 875 y Fw(=)g Fu(K)7 b Fw(,)p 1205 795 V 33 w Fu(M)1310 890 y Fp(3)1377 875 y Fw(=)28 b Fu(M)1575 890 y Fp(3)1642 875 y Fw(=)g Fr(f)p Fw(0)p Fr(g)p Fw(,)k(and)p 1352 1014 V 1352 1094 a Fu(M)1457 1109 y Fp(2)1524 1094 y Fw(=)c(k)m(er)18 b Fu(\013)28 b Fw(=)f Fu(K)j Fr(\001)2131 954 y Fm(\022)2243 1035 y Fw(1)2204 1151 y Fr(\000)p Fw(1)2330 954 y Fm(\023)94 1327 y Fw(\(this)40 b(is)f(the)i(third)e(term)h(in)f(the)h(decomp)s (osition)e(of)p 2168 1247 V 40 w Fu(M)2273 1342 y Fq(k)2355 1327 y Fw(in)h(\()2514 1327 y SDict begin H.S end 2514 1327 a Fw(10.7)2688 1263 y SDict begin H.R end 2688 1263 a 2688 1327 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.7) cvn H.B /ANN pdfmark end 2688 1327 a Fw(\)\).)65 b(Since)40 b Fu(M)3212 1342 y Fp(3)3292 1327 y Fw(=)g(0,)i(the)94 1448 y(arro)m(ws)c Fu(b)452 1412 y Fq(?)452 1473 y Fp(1)492 1448 y Fu(;)17 b(b)577 1412 y Fq(?)577 1473 y Fp(2)617 1448 y Fu(;)g Fw([)p Fu(b)729 1463 y Fp(1)769 1448 y Fu(a)820 1463 y Fp(2)859 1448 y Fw(])p Fu(;)g Fw([)p Fu(b)998 1463 y Fp(2)1038 1448 y Fu(a)1089 1463 y Fp(1)1129 1448 y Fw(])37 b(act)h(as)f(0)g(in)p 1690 1368 105 4 v 37 w Fu(M)11 b Fw(.)57 b(As)38 b(for)f Fu(a)2233 1412 y Fq(?)2233 1473 y Fp(1)2310 1448 y Fw(and)h Fu(a)2556 1412 y Fq(?)2556 1473 y Fp(2)2595 1448 y Fw(,)h(their)e(action)f(is)h (giv)m(en)g(b)m(y)94 1564 y(the)28 b(second)h(equalit)m(y)e(in)f(\() 1079 1564 y SDict begin H.S end 1079 1564 a Fw(10.10)1301 1500 y SDict begin H.R end 1301 1500 a 1301 1564 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.10) cvn H.B /ANN pdfmark end 1301 1564 a Fw(\))i(\(note)f(that)g(the)h(c)m(hoice)g(of) e(a)h(splitting)e(\()2869 1564 y SDict begin H.S end 2869 1564 a Fw(10.9)3043 1500 y SDict begin H.R end 3043 1500 a 3043 1564 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.9) cvn H.B /ANN pdfmark end 3043 1564 a Fw(\))i(is)g(immaterial)94 1742 y(here\).)73 b(Namely)-8 b(,)43 b(iden)m(tifying)p 1292 1662 105 4 v 40 w Fu(M)1397 1757 y Fp(2)1479 1742 y Fw(with)e Fu(K)49 b Fw(via)41 b(c)m(ho)s(osing)2417 1602 y Fm(\022)2529 1682 y Fw(1)2490 1799 y Fr(\000)p Fw(1)2616 1602 y Fm(\023)2732 1742 y Fw(as)h(the)g(standard)g(basis)94 1914 y(v)m(ector,)34 b(w)m(e)g(obtain)1343 2031 y(\()p Fu(a)1432 1990 y Fq(?)1432 2056 y Fp(1)1472 2031 y Fw(\))p 1510 2003 79 3 v 26 x Fq(M)1616 2031 y Fw(=)28 b(1)p Fu(;)114 b Fw(\()p Fu(a)1999 1990 y Fq(?)1999 2056 y Fp(2)2038 2031 y Fw(\))p 2076 2003 V 26 x Fq(M)2183 2031 y Fw(=)28 b Fr(\000)p Fw(1)94 2175 y(as)33 b(maps)p 470 2095 105 4 v 33 w Fu(M)574 2190 y Fp(2)641 2175 y Fw(=)28 b Fu(K)35 b Fr(!)27 b Fu(K)35 b Fw(=)p 1211 2095 V 27 w Fu(M)1316 2190 y Fp(1)1356 2175 y Fw(.)194 2291 y(Note)44 b(that)f(the)h(resulting)f(represen)m(tation)i Fu(\026)1963 2306 y Fp(2)2002 2291 y Fw(\()p Fu(M)10 b Fw(\))44 b(can)g(b)s(e)g(con) m(v)m(enien)m(tly)h(describ)s(ed)f(as)94 2407 y(follo)m(ws:)f(b)m(y)33 b(ren)m(um)m(b)s(ering)g(the)g(v)m(ertices)g(of)f(our)h(quiv)m(er)g (via)94 2565 y(\(11.3\))1345 2565 y SDict begin H.S end 1345 2565 a 1345 2565 a SDict begin 14 H.A end 1345 2565 a 1345 2565 a SDict begin [ /View [/XYZ H.V] /Dest (equation.11.3) cvn H.B /DEST pdfmark end 1345 2565 a Fw(1)1394 2524 y Fn(0)1445 2565 y Fw(=)27 b(2)p Fu(;)114 b Fw(2)1787 2524 y Fn(0)1838 2565 y Fw(=)28 b(1)p Fu(;)114 b Fw(3)2181 2524 y Fn(0)2231 2565 y Fw(=)28 b(3)p Fu(;)94 2723 y Fw(and)33 b(setting)94 2881 y(\(11.4\))675 2881 y SDict begin H.S end 675 2881 a 675 2881 a SDict begin 14 H.A end 675 2881 a 675 2881 a SDict begin [ /View [/XYZ H.V] /Dest (equation.11.4) cvn H.B /DEST pdfmark end 675 2881 a Fu(a)726 2839 y Fn(0)726 2905 y Fp(1)794 2881 y Fw(=)27 b Fr(\000)p Fu(a)1025 2839 y Fq(?)1025 2905 y Fp(2)1065 2881 y Fu(;)50 b(a)1193 2839 y Fn(0)1193 2905 y Fp(2)1261 2881 y Fw(=)27 b Fu(a)1415 2839 y Fq(?)1415 2905 y Fp(1)1455 2881 y Fu(;)50 b(b)1573 2839 y Fn(0)1573 2905 y Fp(1)1640 2881 y Fw(=)28 b([)p Fu(b)1812 2896 y Fp(1)1852 2881 y Fu(a)1903 2896 y Fp(2)1942 2881 y Fw(])p Fu(;)50 b(b)2087 2839 y Fn(0)2087 2905 y Fp(2)2155 2881 y Fw(=)28 b([)p Fu(b)2327 2896 y Fp(2)2367 2881 y Fu(a)2418 2896 y Fp(1)2457 2881 y Fw(])p Fu(;)50 b(c)2603 2839 y Fn(0)2603 2905 y Fp(1)2670 2881 y Fw(=)28 b Fr(\000)p Fu(b)2892 2839 y Fq(?)2892 2905 y Fp(1)2932 2881 y Fu(;)50 b(c)3051 2839 y Fn(0)3051 2905 y Fp(2)3118 2881 y Fw(=)28 b Fu(b)3263 2839 y Fq(?)3263 2905 y Fp(2)3303 2881 y Fu(;)94 3038 y Fw(the)45 b(represen)m(tation)h Fu(\026)983 3053 y Fp(2)1022 3038 y Fw(\()p Fu(M)10 b Fw(\))45 b(gets)g(iden)m(ti\014ed)f(with)g(the)h(initial)c(represen)m (tation)k Fu(M)55 b Fw(of)44 b(the)94 3154 y(initial)29 b(QP)k(\()p Fu(A;)17 b(S)6 b Fw(\).)194 3271 y(The)27 b(m)m(utation)e Fu(\026)861 3286 y Fp(1)900 3271 y Fw(\()p Fu(M)10 b Fw(\))28 b(can)e(b)s(e)h(computed)g(in)e(a)h(similar)e(w)m(a) m(y)-8 b(.)42 b(But)27 b(since)g(w)m(e)g(ha)m(v)m(e)h(already)94 3392 y(computed)41 b(the)g(QP)g Fu(\026)970 3407 y Fp(2)1009 3392 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\))40 b(=)h(\()p 1463 3312 74 4 v Fu(A;)p 1580 3312 66 4 v 17 w(S)6 b Fw(\),)42 b(w)m(e)g(\014nd)e(it)g(more)g(con)m(v)m(enien)m(t)i(to)e(ren)m(um)m(b) s(er)h(the)94 3508 y(v)m(ertices)35 b(via)e(1)658 3472 y Fn(0)710 3508 y Fw(=)28 b(3,)34 b(2)973 3472 y Fn(0)1025 3508 y Fw(=)29 b(1,)k(3)1288 3472 y Fn(0)1340 3508 y Fw(=)c(2,)k(so)g(that)h Fu(\026)1946 3523 y Fp(1)1985 3508 y Fw(\()p Fu(M)10 b Fw(\))34 b(gets)f(iden)m(ti\014ed)h(with)f Fu(\026)3109 3523 y Fp(2)3148 3508 y Fw(\()p Fu(M)3290 3472 y Fn(0)3314 3508 y Fw(\),)g(where)94 3624 y Fu(M)198 3588 y Fn(0)255 3624 y Fw(is)f(giv)m(en)h(b)m(y:)94 3782 y(\(11.5\))823 3782 y SDict begin H.S end 823 3782 a 823 3782 a SDict begin 14 H.A end 823 3782 a 823 3782 a SDict begin [ /View [/XYZ H.V] /Dest (equation.11.5) cvn H.B /DEST pdfmark end 823 3782 a Fu(M)927 3741 y Fn(0)917 3807 y Fp(1)985 3782 y Fw(=)27 b(0)p Fu(;)114 b(M)1382 3741 y Fn(0)1372 3807 y Fp(2)1440 3782 y Fw(=)27 b Fu(M)1647 3741 y Fn(0)1637 3807 y Fp(3)1705 3782 y Fw(=)g Fu(K)r(;)17 b Fw(;)114 b(\()p Fu(b)2157 3797 y Fp(1)2197 3782 y Fw(\))2235 3797 y Fq(M)2310 3778 y Ff(0)2364 3782 y Fw(=)27 b(\()p Fu(b)2546 3797 y Fp(2)2586 3782 y Fw(\))2624 3797 y Fq(M)2699 3778 y Ff(0)2753 3782 y Fw(=)h(1)p Fu(:)94 3940 y Fw(No)m(w)34 b(the)f(triangle)d(\()880 3940 y SDict begin H.S end 880 3940 a Fw(10.5)1054 3876 y SDict begin H.R end 1054 3876 a 1054 3940 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.5) cvn H.B /ANN pdfmark end 1054 3940 a Fw(\))i(is)g(giv)m(en)h(b)m (y)811 4159 y Fu(M)915 4118 y Fn(0)905 4184 y Fp(in)996 4159 y Fw(=)27 b Fr(f)p Fw(0)p Fr(g)p Fu(;)50 b(M)1429 4118 y Fn(0)1419 4184 y Fq(k)1489 4159 y Fw(=)28 b Fu(M)1697 4118 y Fn(0)1687 4184 y Fp(2)1754 4159 y Fw(=)g Fu(K)r(;)50 b(M)2124 4118 y Fn(0)2114 4184 y Fp(out)2248 4159 y Fw(=)27 b Fu(K)2441 4118 y Fp(2)2481 4159 y Fu(;)50 b(\014)33 b Fw(=)2750 4019 y Fm(\022)2823 4099 y Fw(1)2823 4216 y(1)2872 4019 y Fm(\023)94 4378 y Fw(\(so)44 b(w)m(e)g(ha)m(v)m(e)h Fu(\013)i Fw(=)e(0)e(and)h Fu(\015)51 b Fw(=)46 b(0\).)75 b(It)43 b(follo)m(ws)g(that)g Fu(\026)2318 4393 y Fp(2)2357 4378 y Fw(\()p Fu(M)2499 4342 y Fn(0)2523 4378 y Fw(\))g(is)g(p)s (ositiv)m(e,)i(and)f(w)m(e)g(ha)m(v)m(e)p 94 4419 129 4 v 94 4500 a Fu(M)198 4471 y Fn(0)223 4515 y Fp(1)290 4500 y Fw(=)27 b Fu(M)497 4463 y Fn(0)487 4524 y Fp(1)555 4500 y Fw(=)g Fr(f)p Fw(0)p Fr(g)p Fw(,)p 866 4419 V 32 w Fu(M)970 4471 y Fn(0)994 4515 y Fp(3)1061 4500 y Fw(=)h Fu(M)1269 4463 y Fn(0)1259 4524 y Fp(3)1326 4500 y Fw(=)g Fu(K)7 b Fw(,)33 b(and)p 1461 4639 V 1461 4719 a Fu(M)1565 4690 y Fn(0)1589 4734 y Fp(2)1656 4719 y Fw(=)27 b Fu(K)1849 4678 y Fp(2)1889 4719 y Fu(=K)i Fr(\001)2100 4578 y Fm(\022)2173 4659 y Fw(1)2173 4775 y(1)2222 4578 y Fm(\023)94 4951 y Fw(\(this)42 b(is)g(the)g(\014rst)h(term)f(in)f (the)i(decomp)s(osition)d(of)p 2145 4871 105 4 v 42 w Fu(M)2249 4966 y Fq(k)2334 4951 y Fw(in)h(\()2495 4951 y SDict begin H.S end 2495 4951 a Fw(10.7)2669 4887 y SDict begin H.R end 2669 4887 a 2669 4951 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.7) cvn H.B /ANN pdfmark end 2669 4951 a Fw(\)\).)72 b(Since)42 b Fu(M)3212 4915 y Fn(0)3202 4976 y Fp(1)3286 4951 y Fw(=)i(0,)g(the)94 5072 y(arro)m(ws)37 b Fu(a)461 5036 y Fq(?)461 5097 y Fp(1)501 5072 y Fu(;)17 b(a)596 5036 y Fq(?)596 5097 y Fp(2)635 5072 y Fu(;)g Fw([)p Fu(b)747 5087 y Fp(1)787 5072 y Fu(a)838 5087 y Fp(2)878 5072 y Fw(])p Fu(;)g Fw([)p Fu(b)1017 5087 y Fp(2)1056 5072 y Fu(a)1107 5087 y Fp(1)1147 5072 y Fw(])36 b(act)g(as)g(0)g(in)p 1702 4992 129 4 v 36 w Fu(M)1806 5044 y Fn(0)1830 5072 y Fw(.)54 b(As)36 b(for)g Fu(b)2252 5036 y Fq(?)2252 5097 y Fp(1)2328 5072 y Fw(and)g Fu(b)2562 5036 y Fq(?)2562 5097 y Fp(2)2602 5072 y Fw(,)h(their)e(action)h(is)f(giv)m(en)h(b)m(y)94 5189 y(the)g(\014rst)g(equalit)m(y)f(in)g(\()998 5189 y SDict begin H.S end 998 5189 a Fw(10.10)1220 5124 y SDict begin H.R end 1220 5124 a 1220 5189 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.10) cvn H.B /ANN pdfmark end 1220 5189 a Fw(\))g(\(note)h(that)f(the)h(c)m(hoice)g(of)f(a)g (splitting)e(\()2853 5189 y SDict begin H.S end 2853 5189 a Fw(10.8)3027 5124 y SDict begin H.R end 3027 5124 a 3027 5189 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.10.8) cvn H.B /ANN pdfmark end 3027 5189 a Fw(\))i(is)g(immaterial)94 5366 y(here\).)75 b(Namely)-8 b(,)45 b(iden)m(tifying)p 1297 5286 129 4 v 41 w Fu(M)1401 5338 y Fn(0)1425 5381 y Fp(2)1507 5366 y Fw(with)e Fu(K)50 b Fw(via)42 b(c)m(ho)s(osing)g Fu(\031)t Fw(\()2546 5226 y Fm(\022)2619 5307 y Fw(1)2619 5423 y(0)2668 5226 y Fm(\023)2741 5366 y Fw(\))j(=)g Fr(\000)p Fu(\031)t Fw(\()3119 5226 y Fm(\022)3193 5307 y Fw(0)3193 5423 y(1)3242 5226 y Fm(\023)3315 5366 y Fw(\))e(as)g(the)94 5538 y(standard)33 b(basis)g(v)m(ector,)h(w)m(e)f (obtain)1330 5696 y(\()p Fu(b)1409 5655 y Fq(?)1409 5721 y Fp(1)1449 5696 y Fw(\))p 1487 5668 102 3 v 27 x Fq(M)1562 5704 y Ff(0)1616 5696 y Fw(=)28 b Fr(\000)p Fw(1)p Fu(;)114 b Fw(\()p Fu(b)2066 5655 y Fq(?)2066 5721 y Fp(2)2106 5696 y Fw(\))p 2144 5668 V 27 x Fq(M)2219 5704 y Ff(0)2273 5696 y Fw(=)28 b(1)p eop end end %%Page: 53 53 TeXDict begin HPSdict begin 53 52 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.53) cvn H.B /DEST pdfmark end 94 218 a 1179 w Fs(QUIVERS)31 b(WITH)i(POTENTIALS)g(I)1102 b(53)94 422 y Fw(as)33 b(maps)p 470 342 129 4 v 33 w Fu(M)574 393 y Fn(0)598 437 y Fp(3)665 422 y Fw(=)27 b Fu(K)35 b Fr(!)27 b Fu(K)35 b Fw(=)p 1235 342 V 28 w Fu(M)1339 393 y Fn(0)1363 437 y Fp(2)1402 422 y Fw(.)194 538 y(As)e(ab)s(o)m(v)m (e,)h(b)m(y)f(ren)m(um)m(b)s(ering)g(the)g(v)m(ertices)g(of)f(our)h (quiv)m(er)g(via)94 700 y(\(11.6\))1345 700 y SDict begin H.S end 1345 700 a 1345 700 a SDict begin 14 H.A end 1345 700 a 1345 700 a SDict begin [ /View [/XYZ H.V] /Dest (equation.11.6) cvn H.B /DEST pdfmark end 1345 700 a Fw(1)1394 659 y Fn(0)1445 700 y Fw(=)27 b(1)p Fu(;)114 b Fw(2)1787 659 y Fn(0)1838 700 y Fw(=)28 b(3)p Fu(;)114 b Fw(3)2181 659 y Fn(0)2231 700 y Fw(=)28 b(2)p Fu(;)94 863 y Fw(and)33 b(setting)94 1025 y(\(11.7\))675 1025 y SDict begin H.S end 675 1025 a 675 1025 a SDict begin 14 H.A end 675 1025 a 675 1025 a SDict begin [ /View [/XYZ H.V] /Dest (equation.11.7) cvn H.B /DEST pdfmark end 675 1025 a Fu(a)726 984 y Fn(0)726 1050 y Fp(1)794 1025 y Fw(=)27 b([)p Fu(b)965 1040 y Fp(2)1005 1025 y Fu(a)1056 1040 y Fp(1)1096 1025 y Fw(])p Fu(;)50 b(a)1251 984 y Fn(0)1251 1050 y Fp(2)1318 1025 y Fw(=)28 b([)p Fu(b)1490 1040 y Fp(1)1530 1025 y Fu(a)1581 1040 y Fp(2)1620 1025 y Fw(])p Fu(;)50 b(b)1765 984 y Fn(0)1765 1050 y Fp(1)1833 1025 y Fw(=)28 b Fu(b)1978 984 y Fq(?)1978 1050 y Fp(2)2017 1025 y Fu(;)50 b(b)2135 984 y Fn(0)2135 1050 y Fp(2)2203 1025 y Fw(=)27 b Fr(\000)p Fu(b)2424 984 y Fq(?)2424 1050 y Fp(1)2465 1025 y Fu(;)50 b(c)2584 984 y Fn(0)2584 1050 y Fp(1)2651 1025 y Fw(=)27 b Fu(a)2805 984 y Fq(?)2805 1050 y Fp(1)2845 1025 y Fu(;)50 b(c)2964 984 y Fn(0)2964 1050 y Fp(2)3031 1025 y Fw(=)28 b Fr(\000)p Fu(a)3263 984 y Fq(?)3263 1050 y Fp(2)3303 1025 y Fu(;)94 1187 y Fw(the)f(resulting)e(represen)m(tation)i Fu(\026)1341 1202 y Fp(2)1380 1187 y Fw(\()p Fu(M)1522 1151 y Fn(0)1546 1187 y Fw(\))f(gets)g(iden)m(ti\014ed)g(with)g(the)g (initial)c(represen)m(tation)27 b Fu(M)3610 1151 y Fn(0)3634 1187 y Fw(.)194 1303 y(W)-8 b(e)42 b(no)m(w)h(include)e(the)h(represen) m(tations)h Fu(M)53 b Fw(and)42 b Fu(M)2241 1267 y Fn(0)2306 1303 y Fw(giv)m(en)g(b)m(y)h(\()2753 1303 y SDict begin H.S end 2753 1303 a Fw(11.2)2927 1239 y SDict begin H.R end 2927 1239 a 2927 1303 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.11.2) cvn H.B /ANN pdfmark end 2927 1303 a Fw(\))e(and)h(\()3243 1303 y SDict begin H.S end 3243 1303 a Fw(11.5)3416 1239 y SDict begin H.R end 3416 1239 a 3416 1303 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.11.5) cvn H.B /ANN pdfmark end 3416 1303 a Fw(\))g(in)m(to)94 1420 y(a)h(family)d(of)i(p)s(ositiv)m(e)f(represen)m(tations)j(of)e(\() p Fu(A;)17 b(S)6 b Fw(\))42 b(de\014ned)i(as)e(follo)m(ws:)62 b(for)42 b(ev)m(ery)i(pair)e(of)94 1536 y(nonnegativ)m(e)48 b(in)m(tegers)f(\()p Fu(m;)17 b(n)p Fw(\))53 b Fr(6)p Fw(=)f(\(0)p Fu(;)17 b Fw(0\),)50 b(w)m(e)e(de\014ne)g(the)f(p)s (ositiv)m(e)g(represen)m(tation)h Fu(M)63 b Fw(=)94 1652 y Fu(M)10 b Fw(\()p Fu(m;)17 b(n)p Fw(\))34 b(of)e(\()p Fu(A;)17 b(S)6 b Fw(\))32 b(b)m(y)i(setting)94 1814 y(\(11.8\))1129 1814 y SDict begin H.S end 1129 1814 a 1129 1814 a SDict begin 14 H.A end 1129 1814 a 1129 1814 a SDict begin [ /View [/XYZ H.V] /Dest (equation.11.8) cvn H.B /DEST pdfmark end 1129 1814 a Fu(M)1223 1829 y Fp(1)1291 1814 y Fw(=)27 b Fu(K)1484 1773 y Fq(m)1551 1814 y Fu(;)50 b(M)1722 1829 y Fp(2)1790 1814 y Fw(=)27 b Fu(K)1983 1773 y Fq(m)p Fp(+)p Fq(n)2147 1814 y Fu(;)50 b(M)2318 1829 y Fp(3)2386 1814 y Fw(=)27 b Fu(K)2579 1773 y Fq(n)94 1977 y Fw(and)725 2176 y(\()p Fu(a)814 2191 y Fp(1)854 2176 y Fw(\))892 2191 y Fq(M)998 2176 y Fw(=)1102 2035 y Fm(\022)1175 2116 y Fu(I)1218 2131 y Fq(m)1206 2232 y Fw(0)1285 2035 y Fm(\023)1375 2176 y Fu(;)114 b Fw(\()p Fu(a)1605 2191 y Fp(2)1645 2176 y Fw(\))1683 2191 y Fq(M)1789 2176 y Fw(=)1893 2035 y Fm(\022)1997 2116 y Fw(0)1966 2232 y Fu(I)2009 2247 y Fq(m)2076 2035 y Fm(\023)2166 2176 y Fu(;)g Fw(\()p Fu(b)2386 2191 y Fp(1)2426 2176 y Fw(\))2464 2191 y Fq(M)2570 2176 y Fw(=)2674 2095 y Fm(\000)2720 2174 y Fw(0)82 b Fu(I)2894 2189 y Fq(n)2941 2095 y Fm(\001)3004 2176 y Fu(;)94 2176 y SDict begin H.S end 94 2176 a 94 2176 a SDict begin 17.5 H.A end 94 2176 a 94 2176 a SDict begin [ /View [/XYZ H.V] /Dest (equation.11.9) cvn H.B /DEST pdfmark end 94 2176 a Fw(\(11.9\))725 2392 y(\()p Fu(b)804 2407 y Fp(2)844 2392 y Fw(\))882 2407 y Fq(M)989 2392 y Fw(=)1092 2311 y Fm(\000)1138 2390 y Fu(I)1181 2405 y Fq(n)1311 2390 y Fw(0)1360 2311 y Fm(\001)1422 2392 y Fu(;)114 b Fw(\()p Fu(c)1643 2407 y Fp(1)1683 2392 y Fw(\))1721 2407 y Fq(M)1827 2392 y Fw(=)28 b(0)p Fu(;)114 b Fw(\()p Fu(c)2201 2407 y Fp(2)2240 2392 y Fw(\))2278 2407 y Fq(M)2385 2392 y Fw(=)27 b(0)p Fu(;)94 2555 y Fw(where)34 b Fu(I)419 2570 y Fq(n)499 2555 y Fw(is)e(the)h Fu(n)22 b Fr(\002)h Fu(n)33 b Fw(iden)m(tit)m(y)f(matrix.)194 2671 y(W)-8 b(e)36 b(refer)f(to)g(the)h(represen)m(tations)g Fu(M)10 b Fw(\()p Fu(m;)17 b(n)p Fw(\))36 b(as)g(w)m(ell)e(as)i(those)g (obtained)e(from)g(them)h(b)m(y)94 2787 y(ren)m(um)m(b)s(ering)42 b(the)g(v)m(ertices)h(as)f Fv(b)-5 b(and)42 b(r)-5 b(epr)g(esentations) 8 b Fw(;)46 b(they)d(are)e(a)h(sp)s(ecial)e(case)j(of)e(band)94 2903 y(mo)s(dules)34 b(studied)g(in)g([)965 2903 y SDict begin H.S end 965 2903 a Fw(11)1062 2839 y SDict begin H.R end 1062 2839 a 1062 2903 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.BR) cvn H.B /ANN pdfmark end 1062 2903 a Fw(,)1124 2903 y SDict begin H.S end 1124 2903 a Fw(21)1221 2839 y SDict begin H.R end 1221 2839 a 1221 2903 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.F) cvn H.B /ANN pdfmark end 1221 2903 a Fw(])g(in)g(the)g(con)m(text)i(of)e (string)f(algebras.)48 b(Note)34 b(that)g(b)s(oth)g(repre-)94 3020 y(sen)m(tations)29 b Fu(M)39 b Fw(and)29 b Fu(M)972 2983 y Fn(0)1024 3020 y Fw(treated)g(ab)s(o)m(v)m(e)g(are)f(indeed)h (sp)s(ecial)e(cases)j(of)d(band)i(represen)m(tations:)94 3136 y(w)m(e)48 b(ha)m(v)m(e)h Fu(M)62 b Fw(=)52 b Fu(M)10 b Fw(\(1)p Fu(;)17 b Fw(0\),)51 b Fu(M)1280 3100 y Fn(0)1356 3136 y Fw(=)h Fu(M)10 b Fw(\(0)p Fu(;)17 b Fw(1\).)86 b(By)47 b(a)g(direct)f(generalization)f(of)h(the)i(ab)s(o)m(v)m(e)94 3252 y(computations,)32 b(w)m(e)i(obtain)d(the)i(follo)m(wing)d(prop)s (osition.)94 3271 y SDict begin H.S end 94 3271 a 94 3271 a SDict begin 14 H.A end 94 3271 a 94 3271 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.11.4) cvn H.B /DEST pdfmark end 94 3271 a 159 x FA(Prop)s(osition)36 b(11.4.)94 3472 y SDict begin H.S end 94 3472 a 94 3472 a SDict begin 14 H.A end 94 3472 a 94 3472 a SDict begin [ /View [/XYZ H.V] /Dest (Item.9) cvn H.B /DEST pdfmark end 94 3472 a 243 3569 a Fw(\(1\))41 b Fv(If)e Fu(m)e Fr(\025)g Fu(n)j Fv(then)g(after)g(r)-5 b(enumb)g(ering)39 b(of)g(vertic)-5 b(es)39 b(as)h(in)f Fw(\()2641 3569 y SDict begin H.S end 2641 3569 a Fw(11.3)2814 3505 y SDict begin H.R end 2814 3505 a 2814 3569 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.11.3) cvn H.B /ANN pdfmark end 2814 3569 a Fw(\))h Fv(and)f(the)h (change)f(of)409 3685 y(arr)-5 b(ows)45 b(as)g(in)g Fw(\()1039 3685 y SDict begin H.S end 1039 3685 a Fw(11.4)1212 3621 y SDict begin H.R end 1212 3621 a 1212 3685 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.11.4) cvn H.B /ANN pdfmark end 1212 3685 a Fw(\))p Fv(,)j(the)d(r)-5 b(epr)g(esentation)45 b Fu(\026)2205 3700 y Fp(2)2244 3685 y Fw(\()p Fu(M)10 b Fw(\()p Fu(m;)17 b(n)p Fw(\)\))46 b Fv(c)-5 b(an)45 b(b)-5 b(e)45 b(identi\014e)-5 b(d)45 b(with)409 3801 y Fu(M)10 b Fw(\()p Fu(m)23 b Fr(\000)g Fu(n;)17 b(n)p Fw(\))p Fv(.)94 3826 y SDict begin H.S end 94 3826 a 94 3826 a SDict begin 14 H.A end 94 3826 a 94 3826 a SDict begin [ /View [/XYZ H.V] /Dest (Item.10) cvn H.B /DEST pdfmark end 94 3826 a 243 3917 a Fw(\(2\))41 b Fv(If)e Fu(m)e Fr(\024)g Fu(n)j Fv(then)g(after)g(r)-5 b(enumb)g(ering)39 b(of)g(vertic)-5 b(es)39 b(as)h(in)f Fw(\()2641 3917 y SDict begin H.S end 2641 3917 a Fw(11.6)2814 3853 y SDict begin H.R end 2814 3853 a 2814 3917 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.11.6) cvn H.B /ANN pdfmark end 2814 3917 a Fw(\))h Fv(and)f(the)h(change)f(of)409 4034 y(arr)-5 b(ows)45 b(as)g(in)g Fw(\()1039 4034 y SDict begin H.S end 1039 4034 a Fw(11.7)1212 3969 y SDict begin H.R end 1212 3969 a 1212 4034 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.11.7) cvn H.B /ANN pdfmark end 1212 4034 a Fw(\))p Fv(,)j(the)d(r)-5 b(epr)g(esentation)45 b Fu(\026)2205 4049 y Fp(2)2244 4034 y Fw(\()p Fu(M)10 b Fw(\()p Fu(m;)17 b(n)p Fw(\)\))46 b Fv(c)-5 b(an)45 b(b)-5 b(e)45 b(identi\014e)-5 b(d)45 b(with)409 4150 y Fu(M)10 b Fw(\()p Fu(m;)17 b(n)23 b Fr(\000)g Fu(m)p Fw(\))p Fv(.)194 4327 y Fw(Remem)m(b)s(ering)31 b(Theorem)1225 4327 y SDict begin H.S end 1225 4327 a Fw(10.13)1447 4263 y SDict begin H.R end 1447 4263 a 1447 4327 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.10.13) cvn H.B /ANN pdfmark end 1447 4327 a Fw(,)i(w)m(e)h(obtain)d(the)i(follo)m(wing)d (corollary)-8 b(.)94 4347 y SDict begin H.S end 94 4347 a 94 4347 a SDict begin 14 H.A end 94 4347 a 94 4347 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.11.5) cvn H.B /DEST pdfmark end 94 4347 a 158 x FA(Corollary)38 b(11.5.)94 4547 y SDict begin H.S end 94 4547 a 94 4547 a SDict begin 14 H.A end 94 4547 a 94 4547 a SDict begin [ /View [/XYZ H.V] /Dest (Item.11) cvn H.B /DEST pdfmark end 94 4547 a 243 4644 a Fw(\(1\))j Fv(After)j(r)-5 b(enumb)g(ering)42 b(of)g(vertic)-5 b(es)43 b(as)f(in)h Fw(\()2020 4644 y SDict begin H.S end 2020 4644 a Fw(11.3)2193 4580 y SDict begin H.R end 2193 4580 a 2193 4644 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.11.3) cvn H.B /ANN pdfmark end 2193 4644 a Fw(\))p Fv(,)i(the)e(r)-5 b(epr)g(esentation)43 b Fu(\026)3179 4659 y Fp(1)3218 4644 y Fw(\()p Fu(M)10 b Fw(\()p Fu(m;)17 b(n)p Fw(\)\))409 4760 y Fv(b)-5 b(e)g(c)g(omes)34 b(right-e)-5 b(quivalent)34 b(to)h Fu(M)10 b Fw(\()p Fu(m)23 b Fw(+)f Fu(n;)17 b(n)p Fw(\))p Fv(.)94 4785 y SDict begin H.S end 94 4785 a 94 4785 a SDict begin 14 H.A end 94 4785 a 94 4785 a SDict begin [ /View [/XYZ H.V] /Dest (Item.12) cvn H.B /DEST pdfmark end 94 4785 a 243 4876 a Fw(\(2\))41 b Fv(After)j(r)-5 b(enumb)g(ering)42 b(of)g(vertic)-5 b(es)43 b(as)f(in)h Fw(\()2020 4876 y SDict begin H.S end 2020 4876 a Fw(11.6)2193 4812 y SDict begin H.R end 2193 4812 a 2193 4876 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.11.6) cvn H.B /ANN pdfmark end 2193 4876 a Fw(\))p Fv(,)i(the)e(r)-5 b(epr)g(esentation)43 b Fu(\026)3179 4891 y Fp(3)3218 4876 y Fw(\()p Fu(M)10 b Fw(\()p Fu(m;)17 b(n)p Fw(\)\))409 4993 y Fv(b)-5 b(e)g(c)g(omes)34 b(right-e)-5 b(quivalent)34 b(to)h Fu(M)10 b Fw(\()p Fu(m;)17 b(m)23 b Fw(+)f Fu(n)p Fw(\))p Fv(.)94 5079 y SDict begin H.S end 94 5079 a 94 5079 a SDict begin 14 H.A end 94 5079 a 94 5079 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.11.6) cvn H.B /DEST pdfmark end 94 5079 a 91 x FA(Corollary)38 b(11.6.)k Fv(The)34 b(class)g(of)h(b)-5 b(and)34 b(r)-5 b(epr)g(esentations)34 b(is)g(close)-5 b(d)34 b(under)h(mutations.)194 5347 y Fw(Note)44 b(that)f(if)g(w)m(e)i (iterate)e(the)h(m)m(utations)e(in)h(Prop)s(osition)2548 5347 y SDict begin H.S end 2548 5347 a Fw(11.4)2722 5283 y SDict begin H.R end 2722 5283 a 2722 5347 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.11.4) cvn H.B /ANN pdfmark end 2722 5347 a Fw(,)j(the)e(pair)f(\()p Fu(m;)17 b(n)p Fw(\))44 b(gets)94 5464 y(transformed)36 b(according)f(to)g(the)h (Euclid)e(algorithm)f(for)i(\014nding)g(gcd\()p Fu(m;)17 b(n)p Fw(\).)52 b(Th)m(us,)38 b(after)e(a)94 5580 y(sequence)44 b(of)d(m)m(utations)g(\(and)g(appropriate)f(ren)m(um)m(b)s(erings)i(of) f(v)m(ertices\),)j(ev)m(ery)g Fu(M)10 b Fw(\()p Fu(m;)17 b(n)p Fw(\))94 5696 y(can)37 b(b)s(e)e(transformed)h(in)m(to)f Fu(M)10 b Fw(\(gcd)q(\()p Fu(m;)17 b(n)p Fw(\))p Fu(;)g Fw(0\).)53 b(Since)35 b Fu(M)10 b Fw(\()p Fu(d;)17 b Fw(0\))36 b(is)f(ob)m(viously)h(isomorphic)e(to)p eop end end %%Page: 54 54 TeXDict begin HPSdict begin 54 53 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.54) cvn H.B /DEST pdfmark end 94 218 a Fs(54)492 b(HARM)25 b(DERKSEN,)f(JERZY)i(WEYMAN,)f(AND)f(ANDREI)g(ZELEVINSKY)94 417 y Fw(the)39 b(direct)f(sum)g(of)f Fu(d)h Fw(copies)g(of)g Fu(M)10 b Fw(\(1)p Fu(;)17 b Fw(0\),)39 b(b)m(y)g(bac)m(ktrac)m(king)g (this)f(sequence)j(of)c(m)m(utations,)94 533 y(w)m(e)d(obtain)e(the)h (follo)m(wing)c(w)m(ell-kno)m(wn)k(corollary)-8 b(.)94 553 y SDict begin H.S end 94 553 a 94 553 a SDict begin 14 H.A end 94 553 a 94 553 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.11.7) cvn H.B /DEST pdfmark end 94 553 a 167 x FA(Corollary)50 b(11.7.)d Fv(The)c(r)-5 b(epr)g(esentation)44 b Fu(M)10 b Fw(\()p Fu(m;)17 b(n)p Fw(\))45 b Fv(is)f(inde)-5 b(c)g(omp)g(osable)41 b(if)j(and)g(only)g(if)g Fu(m)94 836 y Fv(and)d Fu(n)h Fv(ar)-5 b(e)42 b(r)-5 b(elatively)41 b(prime.)64 b(F)-7 b(urthermor)i(e,)42 b(if)g Fw(gcd\()p Fu(m;)17 b(n)p Fw(\))40 b(=)g Fu(d)h Fv(then)h Fu(M)10 b Fw(\()p Fu(m;)17 b(n)p Fw(\))42 b Fv(is)g(right-)94 953 y(e)-5 b(quivalent)35 b(to)g(the)g(dir)-5 b(e)g(ct)34 b(sum)h(of)g Fu(d)f Fv(c)-5 b(opies)34 b(of)h Fu(M)10 b Fw(\()p Fu(m=d;)17 b(n=d)p Fw(\))p Fv(.)94 1048 y SDict begin H.S end 94 1048 a 94 1048 a SDict begin 14 H.A end 94 1048 a 94 1048 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.11.8) cvn H.B /DEST pdfmark end 94 1048 a 92 x FA(Remark)25 b(11.8.)34 b Fw(By)23 b(the)f(same)g(metho)s(ds)g(as)g(ab)s(o)m(v)m(e,) k(one)c(can)g(compute)g(all)e(the)j(m)m(utations)e(for)94 1256 y(another)33 b(family)e(of)h(represen)m(tations)i(of)e(the)h(QP)g (\()p Fu(A;)17 b(S)6 b Fw(\))33 b(in)f(Example)2803 1256 y SDict begin H.S end 2803 1256 a Fw(11.3)2976 1192 y SDict begin H.R end 2976 1192 a 2976 1256 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.11.3) cvn H.B /ANN pdfmark end 2976 1256 a Fw(:)44 b Fv(string)35 b(mo)-5 b(dules)94 1372 y Fw(in)m(tro)s(duced)33 b(and)g(studied)g(in)f([)1255 1372 y SDict begin H.S end 1255 1372 a Fw(11)1352 1308 y SDict begin H.R end 1352 1308 a 1352 1372 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.BR) cvn H.B /ANN pdfmark end 1352 1372 a Fw(,)1412 1372 y SDict begin H.S end 1412 1372 a Fw(21)1510 1308 y SDict begin H.R end 1510 1308 a 1510 1372 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (cite.F) cvn H.B /ANN pdfmark end 1510 1372 a Fw(].)94 1542 y SDict begin H.S end 94 1542 a 94 1542 a SDict begin 14 H.A end 94 1542 a 94 1542 a SDict begin [ /View [/XYZ H.V] /Dest (section.12) cvn H.B /DEST pdfmark end 94 1542 a 1278 1633 a Fw(12.)49 b Fx(Some)37 b(open)h(pr)n(oblems)194 1807 y Fw(Here)g(w)m(e)g(collect)d(some)i(natural)f(questions)i(that)e (w)m(e)i(\014nd)g(imp)s(ortan)m(t)c(for)j(b)s(etter)g(under-)94 1924 y(standing)31 b(of)g(QPs)g(and)g(their)g(represen)m(tations.)44 b(In)31 b(what)h(follo)m(ws,)e(supp)s(ose)i(that)f(\()p Fu(A;)17 b(S)6 b Fw(\))30 b(is)h(a)94 2040 y(reduced)39 b(QP)e(with)f(the)h(Jacobian)g(algebra)e Fr(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\).)56 b(Let)37 b Fr(M)p Fw(\()p Fu(A;)17 b(S)6 b Fw(\))36 b(denote)i(the)f(category)94 2156 y(of)f Fv(nilp)-5 b(otent)46 b Fw(\014nite)36 b(dimensional)e Fr(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\)-mo)s(dules;)37 b(here,)h(a)e(mo)s(dule)f(is)h(nilp)s(oten)m(t)f(if)g(it)h(is)94 2272 y(annihilated)c(b)m(y)j Fl(m)818 2236 y Fq(n)900 2272 y Fw(for)e Fu(n)e Fr(\035)f Fw(0.)47 b(Supp)s(ose)36 b(also)d(that)h Fu(k)f Fr(2)e Fu(Q)2445 2287 y Fp(0)2518 2272 y Fw(is)j(a)g(v)m(ertex)i(satisfying)d(\()3472 2272 y SDict begin H.S end 3472 2272 a Fw(5.1)3596 2208 y SDict begin H.R end 3596 2208 a 3596 2272 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (equation.5.1) cvn H.B /ANN pdfmark end 3596 2272 a Fw(\),)94 2388 y(so)g(that)g(the)g(m)m(utated)f(reduced)i (QP)f Fu(\026)1578 2403 y Fq(k)1620 2388 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\))32 b(is)g(w)m(ell-de\014ned.)94 2413 y SDict begin H.S end 94 2413 a 94 2413 a SDict begin 14 H.A end 94 2413 a 94 2413 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.12.1) cvn H.B /DEST pdfmark end 94 2413 a 162 x FA(Question)42 b(12.1.)i Fw(Is)37 b(the)f(righ)m(t-equiv)-5 b(alence)36 b(class)g(of)g(\()p Fu(A;)17 b(S)6 b Fw(\))36 b(determined)g(b)m(y)h(the)g(isomor-)94 2692 y(phism)32 b(class)h(of)f(the)h(Jacobian)f(algebra)f Fr(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\)?)94 2787 y SDict begin H.S end 94 2787 a 94 2787 a SDict begin 14 H.A end 94 2787 a 94 2787 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.12.2) cvn H.B /DEST pdfmark end 94 2787 a 92 x FA(Question)35 b(12.2.)40 b Fw(Is)31 b(the)g(isomorphism)d(class)i(of)g Fr(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\))30 b(determined)g(b)m(y)h(the)g(equiv) -5 b(alence)94 2995 y(class)33 b(of)f(the)h(category)g Fr(M)p Fw(\()p Fu(A;)17 b(S)6 b Fw(\)?)94 3091 y SDict begin H.S end 94 3091 a 94 3091 a SDict begin 14 H.A end 94 3091 a 94 3091 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.12.3) cvn H.B /DEST pdfmark end 94 3091 a 91 x FA(Question)34 b(12.3.)39 b Fw(Is)30 b(the)g(isomorphism)d(class)j(of)f Fr(P)8 b Fw(\()p Fu(\026)2175 3197 y Fq(k)2217 3182 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\)\))29 b(determined)g(b)m(y)i(the)f (isomor-)94 3298 y(phism)i(class)h(of)f Fr(P)8 b Fw(\()p Fu(A;)17 b(S)6 b Fw(\)?)94 3394 y SDict begin H.S end 94 3394 a 94 3394 a SDict begin 14 H.A end 94 3394 a 94 3394 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.12.4) cvn H.B /DEST pdfmark end 94 3394 a 91 x FA(Question)46 b(12.4.)f Fw(Is)c(the)f(equiv)-5 b(alence)40 b(class)f(of)g(the)i(category)f Fr(M)p Fw(\()p Fu(\026)2807 3500 y Fq(k)2848 3485 y Fw(\()p Fu(A;)17 b(S)6 b Fw(\)\))40 b(determined)94 3601 y(b)m(y)34 b(the)f(equiv)-5 b(alence)33 b(class)g(of)f Fr(M)p Fw(\()p Fu(A;)17 b(S)6 b Fw(\)?)194 3788 y(W)-8 b(e)33 b(conclude)g(with)f(the) h(follo)m(wing)d(in)m(triguing)g(question.)94 3808 y SDict begin H.S end 94 3808 a 94 3808 a SDict begin 14 H.A end 94 3808 a 94 3808 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.12.5) cvn H.B /DEST pdfmark end 94 3808 a 167 x FA(Question)36 b(12.5.)k Fw(Is)31 b(there)h(a)e(prop)s(er)h(analogue) f(of)g(the)h(cluster)g(category)h(for)e(a)g(non-acyclic)94 4092 y(quiv)m(er)k(with)e(p)s(oten)m(tial?)94 4256 y SDict begin H.S end 94 4256 a 94 4256 a SDict begin 14 H.A end 94 4256 a 94 4256 a SDict begin [ /View [/XYZ H.V] /Dest (section.13) cvn H.B /DEST pdfmark end 94 4256 a 984 4352 a Fw(13.)48 b Fx(Appendix.)g(Pr)n(oof)39 b(of)f(Lemma)2575 4352 y SDict begin H.S end 2575 4352 a Fx(4.12)2772 4288 y SDict begin H.R end 2772 4288 a 2772 4352 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.12) cvn H.B /ANN pdfmark end 2772 4352 a 194 4527 a Fw(W)-8 b(e)32 b(include)f(Lemma)1044 4527 y SDict begin H.S end 1044 4527 a Fw(4.12)1217 4463 y SDict begin H.R end 1217 4463 a 1217 4527 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.12) cvn H.B /ANN pdfmark end 1217 4527 a 32 w Fw(in)m(to)g(a)g(more)g(general)g(setup.)45 b(W)-8 b(e)32 b(call)e(a)h(v)m(ector)i Fu(K)7 b Fw(-space)32 b Fu(V)54 b Fw(a)94 4643 y Fu(C)7 b Fv(-sp)-5 b(ac)g(e)41 b Fw(\(for)34 b(the)h(lac)m(k)f(of)g(a)g(b)s(etter)h(term\))e(if)h Fu(V)55 b Fw(has)35 b(an)f(increasing)g(\014ltration)e Fr(f)p Fw(0)p Fr(g)e Fw(=)h Fu(V)3514 4658 y Fp(0)3584 4643 y Fr(\022)94 4759 y Fu(V)151 4774 y Fp(1)218 4759 y Fr(\022)e(\001)17 b(\001)g(\001)45 b Fw(suc)m(h)32 b(that)f(all)e Fu(V)1106 4774 y Fq(n)1183 4759 y Fw(are)i(\014nite)g (dimensional,)d(and)j Fu(V)49 b Fw(=)2552 4685 y Fm(S)2635 4788 y Fq(n)p Fn(\025)p Fp(0)2789 4759 y Fu(V)2846 4774 y Fq(n)2893 4759 y Fw(.)42 b(\(Equiv)-5 b(alen)m(tly)d(,)31 b Fu(V)94 4883 y Fw(is)h(either)g(\014nite)g(dimensional,)e(or)i(it)f (has)h(coun)m(table)h(dimension.\))42 b(The)33 b(class)f(of)g Fu(C)7 b Fw(-spaces)33 b(is)94 4999 y(clearly)f(closed)h(under)h (taking)e(subspaces,)j(quotien)m(t)e(spaces,)h(\014nite)f(direct)g (sums,)g(and)g(\014nite)94 5115 y(tensor)38 b(pro)s(ducts.)58 b(W)-8 b(e)37 b(alw)m(a)m(ys)h(consider)f Fu(C)7 b Fw(-spaces)38 b(equipp)s(ed)g(with)f(discrete)g(top)s(ology;)h(in)94 5231 y(particular,)31 b(this)i(applies)e(to)i(the)g(base)g(\014eld)f Fu(K)7 b Fw(.)194 5347 y(W)-8 b(e)28 b(refer)f(to)g(the)h(dual)e(space) i Fu(V)1398 5311 y Fq(?)1465 5347 y Fw(of)e(a)h Fu(C)7 b Fw(-space)28 b Fu(V)49 b Fw(as)27 b(a)g Fu(D)s Fv(-sp)-5 b(ac)g(e)33 b Fw(\(the)28 b(dual)e(is)h(understo)s(o)s(d)94 5464 y(as)f(the)f(space)h(of)e(all)f(linear)g(forms)h Fu(V)49 b Fr(!)28 b Fu(K)7 b Fw(\).)41 b(Most)25 b(of)g(the)g(prop)s (erties)g(of)f Fu(D)s Fw(-spaces)i(discussed)94 5580 y(b)s(elo)m(w)36 b(are)f(undoubtedly)h(w)m(ell-kno)m(wn;)g(for)f(the)h (con)m(v)m(enience)h(of)e(the)g(reader,)i(w)m(e)f(pro)m(vide)g(a)94 5696 y(self-con)m(tained)c(treatmen)m(t.)94 5696 y SDict begin H.S end 94 5696 a 94 5696 a SDict begin 14 H.A end 94 5696 a 94 5696 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.13.1) cvn H.B /DEST pdfmark end 94 5696 a eop end end %%Page: 55 55 TeXDict begin HPSdict begin 55 54 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.55) cvn H.B /DEST pdfmark end 94 218 a 1179 w Fs(QUIVERS)31 b(WITH)i(POTENTIALS)g(I)1102 b(55)94 417 y FA(Example)46 b(13.1.)g Fw(The)c(complete)e(path)h(algebra)f Fu(R)q Fr(hh)p Fu(A)p Fr(ii)g Fw(can)h(b)s(e)g(naturally)e(view)m(ed)j (as)f(a)94 537 y Fu(D)s Fw(-space)e Fu(V)556 501 y Fq(?)595 537 y Fw(,)h(corresp)s(onding)e(to)g(the)g Fu(C)7 b Fw(-space)39 b Fu(V)59 b Fw(=)37 b Fr(\010)2274 501 y Fn(1)2274 563 y Fq(d)p Fp(=0)2405 537 y Fw(\()p Fu(A)2516 501 y Fq(d)2557 537 y Fw(\))2595 481 y Fq(?)2634 537 y Fw(,)j(and)e(the)h(\014ltration) d(\()p Fu(V)3577 552 y Fq(n)3623 537 y Fw(\))94 653 y(giv)m(en)d(b)m(y) 1318 773 y Fu(V)1375 788 y Fq(n)1450 773 y Fw(=)28 b Fr(\010)1631 732 y Fq(n)p Fn(\000)p Fp(1)1631 801 y Fq(d)p Fp(=0)1768 773 y Fw(\()p Fu(A)1879 732 y Fq(d)1920 773 y Fw(\))1958 712 y Fq(?)2095 773 y Fw(\()p Fu(n)f Fr(\025)i Fw(1\))p Fu(:)194 947 y Fw(F)-8 b(or)32 b(a)g(subspace)j Fu(W)46 b Fw(of)32 b Fu(V)22 b Fw(,)32 b(w)m(e)i(denote)f(b)m(y)h Fu(W)1945 911 y Fn(?)2031 947 y Fr(\032)28 b Fu(V)2215 911 y Fq(?)2287 947 y Fw(its)k(orthogonal)f(complemen)m(t,)h(that)94 1064 y(is,)1266 1180 y Fu(W)1372 1139 y Fn(?)1458 1180 y Fw(=)c Fr(f)p Fu(f)38 b Fr(2)28 b Fu(V)1871 1139 y Fq(?)1938 1180 y Fr(j)f Fu(f)11 b Fw(\()p Fu(W)j Fw(\))27 b(=)g(0)p Fr(g)p Fu(:)94 1316 y Fw(W)-8 b(e)43 b(mak)m(e)f Fu(V)615 1280 y Fq(?)696 1316 y Fw(in)m(to)g(a)g(top)s(ological)c(v)m (ector)43 b(space)g(b)m(y)h(taking)d(the)h(sets)i Fu(V)2984 1280 y Fn(?)2963 1341 y Fq(n)3085 1316 y Fw(for)e(all)e Fu(n)k Fr(\025)h Fw(0)94 1433 y(as)40 b(a)f(basic)g(system)h(of)e(op)s (en)i(neigh)m(b)s(orho)s(o)s(ds)e(of)h(0.)63 b(In)39 b(particular,)g(in)f(Example)3309 1433 y SDict begin H.S end 3309 1433 a Fw(13.1)3483 1368 y SDict begin H.R end 3483 1368 a 3483 1433 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.13.1) cvn H.B /ANN pdfmark end 3483 1433 a Fw(,)i(w)m(e)94 1549 y(ha)m(v)m(e)d Fu(V)400 1513 y Fn(?)379 1574 y Fq(n)491 1549 y Fw(=)31 b Fl(m)p Fw(\()p Fu(A)p Fw(\))823 1513 y Fq(n)871 1549 y Fw(,)k(so)g(the)g Fu(D)s Fw(-space)g(top)s(ology)f(on)h Fu(R)q Fr(hh)p Fu(A)p Fr(ii)e Fw(coincides)i(with)g(the)g(top)s(ology)94 1665 y(in)m(tro)s(duced)e(in)f(Section)1037 1665 y SDict begin H.S end 1037 1665 a Fw(2)1086 1601 y SDict begin H.R end 1086 1601 a 1086 1665 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (section.2) cvn H.B /ANN pdfmark end 1086 1665 a Fw(.)194 1781 y(Since)c(ev)m(ery)i Fu(v)h Fr(2)d Fu(V)50 b Fw(b)s(elongs)27 b(to)h(some)f Fu(V)1732 1796 y Fq(n)1779 1781 y Fw(,)i(a)f(sequence)i Fu(f)2359 1796 y Fp(1)2399 1781 y Fu(;)17 b(f)2491 1796 y Fp(2)2530 1781 y Fu(;)g(:)g(:)g(:)44 b Fw(con)m(v)m(erges)30 b(in)d Fu(V)3352 1745 y Fq(?)3420 1781 y Fw(if)f(and)94 1898 y(only)36 b(if,)f(for)g(ev)m(ery)j Fu(v)e Fr(2)e Fu(V)21 b Fw(,)37 b(the)f(sequence)i(\()p Fu(f)1834 1913 y Fq(k)1877 1898 y Fw(\()p Fu(v)t Fw(\)\))d(stabilizes)f(as)i Fu(k)g Fr(!)d(1)p Fw(.)52 b(This)36 b(implies)d(in)94 2014 y(particular)h (that)h Fu(W)867 1978 y Fn(?)961 2014 y Fw(is)g(a)g(closed)h(subspace)h (of)e Fu(V)2040 1978 y Fq(?)2115 2014 y Fw(for)g(ev)m(ery)i(subspace)g Fu(W)49 b Fw(of)35 b Fu(V)22 b Fw(.)52 b(In)35 b(fact,)94 2130 y(the)e(con)m(v)m(erse)j(is)c(also)f(true.)94 2130 y SDict begin H.S end 94 2130 a 94 2130 a SDict begin 14 H.A end 94 2130 a 94 2130 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.13.2) cvn H.B /DEST pdfmark end 94 2130 a 172 x FA(Lemma)36 b(13.2.)41 b Fv(A)34 b(ve)-5 b(ctor)34 b(subsp)-5 b(ac)g(e)33 b Fu(Z)41 b Fv(of)33 b Fu(V)1844 2266 y Fq(?)1917 2302 y Fv(is)g(close)-5 b(d)33 b(if)h(and)f(only)h(if) f Fu(Z)i Fw(=)27 b Fu(W)3200 2266 y Fn(?)3293 2302 y Fv(for)33 b(some)94 2419 y(subsp)-5 b(ac)g(e)35 b Fu(W)48 b Fv(of)35 b Fu(V)21 b Fv(.)94 2591 y(Pr)-5 b(o)g(of.)42 b Fw(Let)32 b Fu(Z)40 b Fw(b)s(e)33 b(a)f(v)m(ector)i(subspace)g(of)e Fu(V)1778 2555 y Fq(?)1817 2591 y Fw(.)44 b(Let)1128 2748 y Fu(W)e Fw(=)27 b Fr(f)p Fu(v)32 b Fr(2)c Fu(V)49 b Fr(j)27 b Fu(f)11 b Fw(\()p Fu(v)t Fw(\))27 b(=)h(0)k(for)g Fu(f)38 b Fr(2)29 b Fu(Z)7 b Fr(g)p Fw(.)94 2910 y(It)48 b(su\016ces)i(to)d(sho)m(w)h(that)g Fu(W)1285 2874 y Fn(?)1391 2910 y Fw(is)f(con)m(tained)g(in)g(the)h(closure)p 2612 2830 74 4 v 47 w Fu(Z)55 b Fw(of)47 b Fu(Z)7 b Fw(.)88 b(Let)47 b Fu(f)64 b Fr(2)54 b Fu(W)3576 2874 y Fn(?)3634 2910 y Fw(.)94 3026 y(Restricting)22 b Fu(f)34 b Fw(to)23 b(a)g(\014nite-dimensional)c(subspace)25 b Fu(V)2080 3041 y Fq(n)2150 3026 y Fw(of)e Fu(V)e Fw(,)k(w)m(e)g(conclude)e(that)g Fu(f)38 b Fr(j)3224 3041 y Fq(V)3265 3049 y Fg(n)3312 3026 y Fw(=)27 b Fu(h)3471 3041 y Fq(n)3546 3026 y Fr(j)3574 3041 y Fq(V)3615 3049 y Fg(n)94 3147 y Fw(for)e(some)g Fu(h)529 3162 y Fq(n)604 3147 y Fr(2)j Fu(Z)7 b Fw(.)41 b(Th)m(us,)28 b(the)d(sequence)j Fu(h)1721 3162 y Fp(1)1761 3147 y Fu(;)17 b(h)1861 3162 y Fp(2)1900 3147 y Fu(;)g(:)g(:)g(:)f(;)25 b Fw(con)m(v)m(erges)i(to)e Fu(f)11 b Fw(,)26 b(implying)c(that)j Fu(f)38 b Fr(2)p 3560 3067 V 28 w Fu(Z)7 b Fw(,)94 3264 y(as)33 b(required.)2994 b Fh(\003)194 3430 y Fw(In)29 b(view)f(of)g(Lemma)983 3430 y SDict begin H.S end 983 3430 a Fw(13.2)1156 3365 y SDict begin H.R end 1156 3365 a 1156 3430 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.13.2) cvn H.B /ANN pdfmark end 1156 3430 a Fw(,)i(for)d(ev)m(ery)j (closed)f(subspace)h Fu(Z)35 b Fw(of)28 b Fu(V)2585 3393 y Fq(?)2624 3430 y Fw(,)h(the)g(spaces)h Fu(Z)35 b Fw(and)28 b Fu(V)3505 3393 y Fq(?)3544 3430 y Fu(=)-5 b(Z)94 3546 y Fw(can)33 b(b)s(e)g(naturally)e(view)m(ed)j(as)f Fu(D)s Fw(-spaces:)44 b(indeed,)33 b(w)m(e)h(ha)m(v)m(e)901 3703 y Fu(Z)h Fw(=)27 b Fu(W)1212 3662 y Fn(?)1298 3703 y Fw(=)h(\()p Fu(V)11 b(=W)j Fw(\))1701 3662 y Fq(?)1739 3703 y Fu(;)115 b(V)1959 3662 y Fq(?)1999 3703 y Fu(=)-5 b(Z)34 b Fw(=)27 b Fu(V)2326 3662 y Fq(?)2365 3703 y Fu(=W)2520 3662 y Fn(?)2606 3703 y Fw(=)h Fu(W)2816 3662 y Fq(?)94 3860 y Fw(for)33 b(some)f(subspace)i Fu(W)47 b Fw(of)32 b Fu(V)21 b Fw(.)44 b(The)33 b(follo)m(wing)d(lemma)g(is)j (immediate)c(from)j(the)h(de\014nitions.)94 3879 y SDict begin H.S end 94 3879 a 94 3879 a SDict begin 14 H.A end 94 3879 a 94 3879 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.13.3) cvn H.B /DEST pdfmark end 94 3879 a 153 x FA(Lemma)39 b(13.3.)j Fv(F)-7 b(or)35 b(every)h(close)-5 b(d)35 b(subsp)-5 b(ac)g(e)35 b Fu(Z)h Fr(\022)30 b Fu(V)2173 3996 y Fq(?)2212 4032 y Fv(,)36 b(the)g Fu(D)s Fv(-sp)-5 b(ac)g(e)35 b(top)-5 b(olo)g(gies)35 b(on)g Fu(Z)43 b Fv(and)94 4148 y Fu(V)173 4112 y Fq(?)212 4148 y Fu(=)-5 b(Z)52 b Fv(c)-5 b(oincide)43 b(with)i(the)f(top)-5 b(olo)g(gies)44 b(induc)-5 b(e)g(d)44 b(fr)-5 b(om)45 b Fu(V)2287 4112 y Fq(?)2327 4148 y Fv(.)74 b(In)44 b(p)-5 b(articular,)47 b(the)e(emb)-5 b(e)g(dding)94 4265 y Fu(Z)35 b Fr(!)27 b Fu(V)402 4228 y Fq(?)476 4265 y Fv(and)35 b(the)f(pr)-5 b(oje)g(ction)34 b Fu(V)1358 4228 y Fq(?)1425 4265 y Fr(!)27 b Fu(V)1631 4228 y Fq(?)1670 4265 y Fu(=)-5 b(Z)42 b Fv(ar)-5 b(e)34 b(c)-5 b(ontinuous.)94 4346 y SDict begin H.S end 94 4346 a 94 4346 a SDict begin 14 H.A end 94 4346 a 94 4346 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.13.4) cvn H.B /DEST pdfmark end 94 4346 a 91 x FA(Lemma)47 b(13.4.)f Fv(If)c Fu(Z)954 4452 y Fp(1)1035 4437 y Fv(and)g Fu(Z)1299 4452 y Fp(2)1381 4437 y Fv(ar)-5 b(e)42 b(close)-5 b(d)41 b(subsp)-5 b(ac)g(es)42 b(of)g Fu(V)2484 4401 y Fq(?)2524 4437 y Fv(,)i(then)e Fu(Z)2889 4452 y Fp(1)2956 4437 y Fw(+)28 b Fu(Z)3127 4452 y Fp(2)3208 4437 y Fv(is)42 b(a)g(close)-5 b(d)94 4553 y(subsp)g(ac)g(e)35 b(of)f Fu(V)679 4517 y Fq(?)753 4553 y Fv(as)g(wel)5 b(l.)94 4726 y(Pr)-5 b(o)g(of.)42 b Fw(By)f(Lemma)911 4726 y SDict begin H.S end 911 4726 a Fw(13.2)1084 4661 y SDict begin H.R end 1084 4661 a 1084 4726 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.13.2) cvn H.B /ANN pdfmark end 1084 4726 a Fw(,)i Fu(Z)1221 4741 y Fp(1)1302 4726 y Fw(=)f Fu(W)1526 4689 y Fn(?)1512 4750 y Fp(1)1625 4726 y Fw(and)g Fu(Z)1891 4741 y Fp(2)1972 4726 y Fw(=)f Fu(W)2195 4689 y Fn(?)2181 4750 y Fp(2)2295 4726 y Fw(for)f(some)h (subspaces)i Fu(W)3251 4741 y Fp(1)3332 4726 y Fw(and)e Fu(W)3622 4741 y Fp(2)94 4842 y Fw(of)c Fu(V)22 b Fw(.)58 b(Cho)s(osing)37 b(some)g(direct)h(complemen)m(ts)f(of)g Fu(W)2132 4857 y Fp(1)2197 4842 y Fr(\\)25 b Fu(W)2380 4857 y Fp(2)2457 4842 y Fw(in)37 b Fu(W)2668 4857 y Fp(1)2745 4842 y Fw(and)h Fu(W)3032 4857 y Fp(2)3071 4842 y Fw(,)h(and)e(a)h (direct)94 4958 y(complemen)m(t)32 b(of)g Fu(W)842 4973 y Fp(1)904 4958 y Fw(+)22 b Fu(W)1094 4973 y Fp(2)1166 4958 y Fw(in)32 b Fu(V)21 b Fw(,)33 b(it)e(is)i(easy)g(to)f(see)i(that) 1088 5115 y Fu(Z)1155 5130 y Fp(1)1216 5115 y Fw(+)22 b Fu(Z)1381 5130 y Fp(2)1448 5115 y Fw(=)28 b Fu(W)1658 5074 y Fn(?)1644 5140 y Fp(1)1738 5115 y Fw(+)22 b Fu(W)1942 5074 y Fn(?)1928 5140 y Fp(2)2029 5115 y Fw(=)27 b(\()p Fu(W)2262 5130 y Fp(1)2324 5115 y Fr(\\)22 b Fu(W)2504 5130 y Fp(2)2544 5115 y Fw(\))2582 5074 y Fn(?)2641 5115 y Fu(;)94 5272 y Fw(pro)m(ving)33 b(that)f Fu(Z)725 5287 y Fp(1)786 5272 y Fw(+)22 b Fu(Z)951 5287 y Fp(2)1023 5272 y Fw(is)32 b(closed.)2181 b Fh(\003)94 5341 y SDict begin H.S end 94 5341 a 94 5341 a SDict begin 14 H.A end 94 5341 a 94 5341 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.13.5) cvn H.B /DEST pdfmark end 94 5341 a 103 x FA(Lemma)43 b(13.5.)h Fv(L)-5 b(et)39 b Fu(U)50 b Fv(and)39 b Fu(V)60 b Fv(b)-5 b(e)39 b Fu(C)7 b Fv(-sp)-5 b(ac)g(es,)39 b(and)f Fu(U)2204 5408 y Fq(?)2283 5444 y Fv(and)h Fu(V)2555 5408 y Fq(?)2634 5444 y Fv(b)-5 b(e)38 b(the)h(c)-5 b(orr)g(esp)g (onding)38 b Fu(D)s Fv(-)94 5561 y(sp)-5 b(ac)g(es.)65 b(A)43 b(line)-5 b(ar)41 b(map)g Fu(\013)g Fw(:)g Fu(U)1314 5524 y Fq(?)1395 5561 y Fr(!)f Fu(V)1614 5524 y Fq(?)1695 5561 y Fv(is)i(c)-5 b(ontinuous)41 b(if)h(and)f(only)h(if)g Fu(\013)f Fw(=)g Fu(\014)3204 5524 y Fq(?)3285 5561 y Fv(for)g(some)94 5677 y(line)-5 b(ar)35 b(map)f Fu(\014)f Fw(:)28 b Fu(V)49 b Fr(!)27 b Fu(U)10 b Fv(.)p eop end end %%Page: 56 56 TeXDict begin HPSdict begin 56 55 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.56) cvn H.B /DEST pdfmark end 94 218 a Fs(56)492 b(HARM)25 b(DERKSEN,)f(JERZY)i(WEYMAN,)f(AND)f(ANDREI)g(ZELEVINSKY)94 417 y Fv(Pr)-5 b(o)g(of.)42 b Fw(First)28 b(let)h(us)h(sho)m(w)h(that)f Fu(\013)e Fw(=)g Fu(\014)1590 381 y Fq(?)1658 417 y Fw(is)h(con)m(tin)m (uous.)43 b(By)31 b(the)f(de\014nition,)f(it)g(is)g(enough)h(to)94 533 y(sho)m(w)37 b(that,)g(for)e(ev)m(ery)j Fu(n)p Fw(,)f(there)g (exists)f(an)g(index)g Fu(k)j Fw(suc)m(h)f(that)d Fu(U)2651 497 y Fn(?)2641 559 y Fq(k)2744 533 y Fr(\032)f Fu(\013)2918 497 y Fn(\000)p Fp(1)3012 533 y Fw(\()p Fu(V)3129 497 y Fn(?)3107 558 y Fq(n)3188 533 y Fw(.)53 b(Since)36 b(the)94 649 y(subspace)j Fu(\014)6 b Fw(\()p Fu(V)662 664 y Fq(n)708 649 y Fw(\))35 b Fr(\032)f Fu(U)47 b Fw(is)36 b(\014nite)h(dimensional,)e(it)g(is)h(con)m(tained)h(in)f(some)g Fu(U)3011 664 y Fq(k)3054 649 y Fw(,)h(implying)d(the)94 766 y(desired)g(inclusion)d Fu(U)909 730 y Fn(?)899 792 y Fq(k)996 766 y Fr(\032)d Fu(\013)1164 730 y Fn(\000)p Fp(1)1258 766 y Fw(\()p Fu(V)1375 730 y Fn(?)1353 790 y Fq(n)1434 766 y Fw(.)194 882 y(Con)m(v)m(ersely)-8 b(,)40 b(supp)s(ose)f Fu(\013)d Fw(:)f Fu(U)1320 846 y Fq(?)1395 882 y Fr(!)g Fu(V)1609 846 y Fq(?)1685 882 y Fw(is)i(a)g(con)m(tin)m(uous)g(linear)f(map.)56 b(Let)37 b Fu(v)i Fr(2)d Fu(V)21 b Fw(.)57 b(Then)94 998 y(the)30 b(linear)e(form)g Fu(f)38 b Fr(7!)27 b Fu(\013)q Fw(\()p Fu(f)11 b Fw(\)\()p Fu(v)t Fw(\))28 b(is)h(a)g(con)m(tin)m(uous)h (linear)d(map)i Fu(U)2535 962 y Fq(?)2602 998 y Fr(!)f Fu(K)7 b Fw(,)30 b(and)f(so)g(its)g(k)m(ernel)h(is)94 1114 y(a)j(closed)f(subspace)i(of)e Fu(U)1057 1078 y Fq(?)1097 1114 y Fw(.)44 b(Using)32 b(Lemma)1790 1114 y SDict begin H.S end 1790 1114 a Fw(13.2)1963 1050 y SDict begin H.R end 1963 1050 a 1963 1114 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.13.2) cvn H.B /ANN pdfmark end 1963 1114 a Fw(,)g(w)m(e)i(conclude)f(that)f(there)h(exists)g(a)f (unique)94 1231 y Fu(u)j Fr(2)g Fu(U)47 b Fw(suc)m(h)39 b(that)d Fu(\013)q Fw(\()p Fu(f)11 b Fw(\)\()p Fu(v)t Fw(\))34 b(=)g Fu(f)11 b Fw(\()p Fu(u)p Fw(\))36 b(for)g(all)f Fu(f)46 b Fr(2)35 b Fu(U)2099 1194 y Fq(?)2139 1231 y Fw(.)56 b(The)38 b(corresp)s(ondence)h Fu(v)f Fr(7!)c Fu(u)j Fw(is)f(the)94 1347 y(desired)e(linear)d(map)g Fu(\014)j Fw(:)27 b Fu(V)50 b Fr(!)27 b Fu(U)43 b Fw(suc)m(h)34 b(that)e Fu(\013)d Fw(=)e Fu(\014)2085 1311 y Fq(?)2124 1347 y Fw(.)1433 b Fh(\003)94 1473 y SDict begin H.S end 94 1473 a 94 1473 a SDict begin 14 H.A end 94 1473 a 94 1473 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.13.6) cvn H.B /DEST pdfmark end 94 1473 a 75 x FA(Lemma)45 b(13.6.)g Fv(A)n(ny)c(c)-5 b(ontinuous)41 b(line)-5 b(ar)40 b(map)g(of)g Fu(D)s Fv(-sp)-5 b(ac)g(es)40 b Fu(\013)f Fw(:)g Fu(U)2767 1512 y Fq(?)2845 1548 y Fr(!)g Fu(V)3062 1512 y Fq(?)3142 1548 y Fv(sends)h(close)-5 b(d)94 1664 y(ve)g(ctor)35 b(subsp)-5 b(ac)g(es)34 b(of)h Fu(U)1004 1628 y Fq(?)1079 1664 y Fv(to)g(close)-5 b(d)34 b(ve)-5 b(ctor)34 b(subsp)-5 b(ac)g(es)34 b(of)h Fu(V)2391 1628 y Fq(?)2431 1664 y Fv(.)94 1865 y(Pr)-5 b(o)g(of.)42 b Fw(Let)32 b Fu(Z)j Fr(\022)28 b Fu(U)850 1829 y Fq(?)923 1865 y Fw(b)s(e)33 b(a)f(closed)h(v)m(ector)h(subspace.)45 b(By)34 b(Lemma)2665 1865 y SDict begin H.S end 2665 1865 a Fw(13.2)2838 1801 y SDict begin H.R end 2838 1801 a 2838 1865 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.13.2) cvn H.B /ANN pdfmark end 2838 1865 a Fw(,)f Fu(Z)i Fw(=)27 b Fu(W)3209 1829 y Fn(?)3300 1865 y Fw(for)33 b(some)94 1981 y(v)m(ector)42 b(subspace)g Fu(W)54 b Fr(\032)40 b Fu(U)10 b Fw(.)67 b(Also)39 b(b)m(y)i(Lemma)1969 1981 y SDict begin H.S end 1969 1981 a Fw(13.5)2142 1917 y SDict begin H.R end 2142 1917 a 2142 1981 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.13.5) cvn H.B /ANN pdfmark end 2142 1981 a Fw(,)h(w)m(e)f(ha)m(v)m(e)g Fu(\013)g Fw(=)g Fu(\014)2875 1945 y Fq(?)2954 1981 y Fw(for)e(a)h(linear)e(map)94 2097 y Fu(\014)g Fw(:)32 b Fu(V)53 b Fr(!)32 b Fu(U)10 b Fw(.)51 b(The)36 b(de\014nitions)f (imply)e(that)h Fu(\013)q Fw(\()p Fu(Z)7 b Fw(\))32 b(=)f Fu(\014)2222 2061 y Fq(?)2261 2097 y Fw(\()p Fu(W)2405 2061 y Fn(?)2464 2097 y Fw(\))h(=)g(\()p Fu(\014)2741 2061 y Fn(\000)p Fp(1)2834 2097 y Fw(\()p Fu(W)14 b Fw(\)\))3054 2061 y Fn(?)3113 2097 y Fw(,)36 b(hence)g Fu(\013)q Fw(\()p Fu(Z)7 b Fw(\))94 2214 y(is)33 b(a)f(closed)h(subspace)h(of)e Fu(V)1159 2177 y Fq(?)1198 2214 y Fw(,)h(as)g(claimed.)1852 b Fh(\003)194 2436 y Fw(W)-8 b(e)49 b(will)c(call)i(a)g Fu(D)s Fw(-space)i Fu(V)1339 2400 y Fq(?)1427 2436 y Fw(a)f Fu(D)s Fv(-algebr)-5 b(a)54 b Fw(if)47 b(it)g(has)h(a)g (structure)h(of)f(an)g(asso)s(ciativ)m(e)94 2553 y Fu(K)7 b Fw(-algebra)42 b(suc)m(h)i(that)f Fu(V)1099 2517 y Fn(?)1078 2577 y Fq(m)1158 2553 y Fu(V)1237 2517 y Fn(?)1215 2577 y Fq(n)1341 2553 y Fr(\032)i Fu(V)1542 2517 y Fn(?)1520 2577 y Fq(m)p Fp(+)p Fq(n)1727 2553 y Fw(for)d(all)f Fu(m;)17 b(n)45 b Fr(\025)h Fw(0.)74 b(In)43 b(particular,)g Fu(R)q Fr(hh)p Fu(A)p Fr(ii)f Fw(is)h(a)94 2669 y Fu(D)s Fw(-algebra.)94 2688 y SDict begin H.S end 94 2688 a 94 2688 a SDict begin 14 H.A end 94 2688 a 94 2688 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.13.7) cvn H.B /DEST pdfmark end 94 2688 a 182 x FA(Lemma)28 b(13.7.)35 b Fv(If)27 b Fu(I)885 2885 y Fp(1)924 2870 y Fu(;)17 b(:)g(:)g(:)f(;)h(I)1186 2885 y Fq(N)1280 2870 y Fv(ar)-5 b(e)27 b(close)-5 b(d)26 b(subsp)-5 b(ac)g(es)26 b(in)g(a)h Fu(D)s Fv(-algebr)-5 b(a)26 b Fu(V)2844 2834 y Fq(?)2884 2870 y Fv(,)i(then)f(the)g(subsp)-5 b(ac)g(e)94 2986 y Fu(I)137 3001 y Fp(1)177 2986 y Fu(f)225 3001 y Fp(1)276 2986 y Fw(+)11 b Fr(\001)17 b(\001)g(\001)10 b Fw(+)h Fu(I)621 3001 y Fq(N)689 2986 y Fu(f)737 3001 y Fq(N)834 2986 y Fv(is)30 b(close)-5 b(d)29 b(for)h(every)g Fu(f)1665 3001 y Fp(1)1704 2986 y Fu(;)17 b(:)g(:)g(:)f(;)h(f)1971 3001 y Fq(N)2066 2986 y Fr(2)28 b Fu(V)2238 2950 y Fq(?)2278 2986 y Fv(.)43 b(In)29 b(p)-5 b(articular,)31 b(\014nitely)f(gener)-5 b(ate)g(d)94 3102 y(left)35 b(ide)-5 b(als)34 b(in)h Fu(V)728 3066 y Fq(?)803 3102 y Fv(ar)-5 b(e)34 b(close)-5 b(d.)94 3303 y(Pr)g(o)g(of.)42 b Fw(By)27 b(the)g(de\014nition)e(of)h (a)h Fu(D)s Fw(-algebra,)f(the)h(op)s(erator)f(of)g(righ)m(t)g(m)m (ultiplication)21 b(with)27 b(an)m(y)94 3419 y Fu(f)48 b Fr(2)36 b Fu(V)371 3383 y Fq(?)448 3419 y Fw(is)h(con)m(tin)m(uous.) 60 b(Th)m(us)39 b(eac)m(h)g(subspace)g Fu(I)2025 3434 y Fq(k)2068 3419 y Fu(f)2116 3434 y Fq(k)2196 3419 y Fw(is)f(closed)f(b)m(y)i(Lemma)3086 3419 y SDict begin H.S end 3086 3419 a Fw(13.6)3260 3355 y SDict begin H.R end 3260 3355 a 3260 3419 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.13.6) cvn H.B /ANN pdfmark end 3260 3419 a Fw(,)g(and)f(our)94 3536 y(assertion)33 b(follo)m(ws)e(from)h(Lemma) 1401 3536 y SDict begin H.S end 1401 3536 a Fw(13.4)1574 3471 y SDict begin H.R end 1574 3471 a 1574 3536 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.13.4) cvn H.B /ANN pdfmark end 1574 3536 a Fw(.)1983 b Fh(\003)194 3758 y Fw(Recall)43 b(from)g(De\014nition)1208 3758 y SDict begin H.S end 1208 3758 a Fw(3.4)1332 3694 y SDict begin H.R end 1332 3694 a 1332 3758 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.3.4) cvn H.B /ANN pdfmark end 1332 3758 a Fw(,)k(that)e(the)f(trace)h(space)g(of)f(a)g Fu(D)s Fw(-algebra)f Fu(V)3101 3722 y Fq(?)3185 3758 y Fw(is)h(the)g(quo-)94 3875 y(tien)m(t)38 b(T)-8 b(r\()p Fu(V)547 3839 y Fq(?)586 3875 y Fw(\))37 b(=)g Fu(V)852 3839 y Fq(?)892 3875 y Fu(=)p Fr(f)p Fu(V)1069 3839 y Fq(?)1108 3875 y Fu(;)17 b(V)1231 3839 y Fq(?)1270 3875 y Fr(g)p Fw(,)39 b(where)g Fr(f)p Fu(V)1802 3839 y Fq(?)1841 3875 y Fu(;)17 b(V)1963 3839 y Fq(?)2003 3875 y Fr(g)38 b Fw(is)f(the)h(closure)h(of)e(the)h(v) m(ector)h(subspace)94 3991 y(in)e Fu(V)292 3955 y Fq(?)369 3991 y Fw(spanned)i(b)m(y)g(all)d(comm)m(utators.)58 b(W)-8 b(e)38 b(denote)g(b)m(y)h Fu(\031)i Fw(:)36 b Fu(V)2549 3955 y Fq(?)2625 3991 y Fr(!)g Fw(T)-8 b(r\()p Fu(V)2978 3955 y Fq(?)3017 3991 y Fw(\))38 b(the)g(canonical)94 4107 y(pro)5 b(jection.)43 b(By)32 b(Lemma)1093 4107 y SDict begin H.S end 1093 4107 a Fw(13.3)1266 4043 y SDict begin H.R end 1266 4043 a 1266 4107 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.13.3) cvn H.B /ANN pdfmark end 1266 4107 a Fw(,)g Fu(\031)i Fw(is)d(con)m(tin)m(uous)g(with)f(resp)s (ect)j(to)d(the)h Fu(D)s Fw(-space)g(top)s(ologies.)194 4223 y(In)40 b(view)g(of)g(Prop)s(osition)1206 4223 y SDict begin H.S end 1206 4223 a Fw(3.5)1330 4159 y SDict begin H.R end 1330 4159 a 1330 4223 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.3.5) cvn H.B /ANN pdfmark end 1330 4223 a Fw(,)i(the)e(assertion)g(of)f(Lemma)2462 4223 y SDict begin H.S end 2462 4223 a Fw(4.12)2636 4159 y SDict begin H.R end 2636 4159 a 2636 4223 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.4.12) cvn H.B /ANN pdfmark end 2636 4223 a 39 w Fw(is)h(a)f(sp)s(ecial)g(case)i(of)e(the)94 4340 y(follo)m(wing.)94 4359 y SDict begin H.S end 94 4359 a 94 4359 a SDict begin 14 H.A end 94 4359 a 94 4359 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.13.8) cvn H.B /DEST pdfmark end 94 4359 a 182 x FA(Lemma)56 b(13.8.)50 b Fv(L)-5 b(et)49 b Fu(I)58 b Fv(b)-5 b(e)49 b(a)g(close)-5 b(d)48 b(\(two-side)-5 b(d\))48 b(ide)-5 b(al)49 b(of)g(a)g Fu(D)s Fv(-algebr)-5 b(a)49 b Fu(V)3138 4504 y Fq(?)3178 4541 y Fv(,)j(and)d Fu(J)59 b Fv(b)-5 b(e)94 4657 y(the)38 b(closur)-5 b(e)37 b(of)g(an)g(ide)-5 b(al)37 b(gener)-5 b(ate)g(d)36 b(by)i(\014nitely)f(many)g(elements)g Fu(f)2691 4672 y Fp(1)2730 4657 y Fu(;)17 b(f)2822 4672 y Fp(2)2862 4657 y Fu(;)g(:)g(:)g(:)e(;)i(f)3128 4672 y Fq(N)3195 4657 y Fv(.)53 b(Then)36 b(the)94 4773 y(subsp)-5 b(ac)g(e)35 b Fu(\031)t Fw(\()p Fu(I)8 b(J)h Fw(\))27 b Fr(\022)h Fw(T)-8 b(r\()p Fu(V)1084 4737 y Fq(?)1123 4773 y Fw(\))35 b Fv(is)g(e)-5 b(qual)34 b(to)h Fu(\031)t Fw(\()p Fu(I)8 b(f)1860 4788 y Fp(1)1922 4773 y Fw(+)22 b Fr(\001)17 b(\001)g(\001)j Fw(+)i Fu(I)8 b(f)2355 4788 y Fq(N)2422 4773 y Fw(\))p Fv(.)94 4974 y(Pr)-5 b(o)g(of.)42 b Fw(Let)i Fu(J)642 4938 y Fp(0)725 4974 y Fw(b)s(e)g(the)g(ideal)e(generated)j(b) m(y)f Fu(f)1942 4989 y Fp(1)1982 4974 y Fu(;)17 b(f)2074 4989 y Fp(2)2113 4974 y Fu(;)g(:)g(:)g(:)f(;)h(f)2380 4989 y Fq(N)2447 4974 y Fw(,)47 b(that)c(is,)j(the)e(linear)f(span)h (of)94 5090 y(elemen)m(ts)e(of)f(the)h(form)e Fu(uf)1142 5105 y Fq(k)1184 5090 y Fu(v)45 b Fw(with)c Fu(u;)17 b(v)45 b Fr(2)e Fu(V)1887 5054 y Fq(?)1968 5090 y Fw(and)e Fu(k)46 b Fw(=)c(1)p Fu(;)17 b(:)g(:)g(:)f(;)h(N)10 b Fw(.)70 b(Th)m(us)42 b(the)g(ideal)e Fu(I)8 b(J)3622 5054 y Fp(0)94 5206 y Fw(is)37 b(the)h(linear)d(span)j(of)f(elemen)m (ts)g(of)g(the)g(form)f Fu(g)t(uf)2075 5221 y Fq(k)2117 5206 y Fu(v)k Fw(with)d Fu(g)i Fr(2)d Fu(I)8 b Fw(.)57 b(By)37 b(the)h(de\014nition,)f(w)m(e)94 5323 y(ha)m(v)m(e)e Fu(\031)t Fw(\()p Fu(g)t(uf)572 5338 y Fq(k)614 5323 y Fu(v)t Fw(\))29 b(=)g Fu(\031)t Fw(\()p Fu(v)t(g)t(uf)1140 5338 y Fq(k)1181 5323 y Fw(\),)34 b(and)g(so)f Fu(\031)t Fw(\()p Fu(I)8 b(J)1802 5286 y Fp(0)1842 5323 y Fw(\))29 b(=)g Fu(\031)t Fw(\()p Fu(I)8 b(f)2210 5338 y Fp(1)2272 5323 y Fw(+)23 b Fr(\001)17 b(\001)g(\001)k Fw(+)i Fu(I)8 b(f)2708 5338 y Fq(N)2775 5323 y Fw(\).)46 b(Since)34 b Fu(I)8 b(J)3256 5286 y Fp(0)3329 5323 y Fw(is)33 b(dense)94 5439 y(in)40 b Fu(I)8 b(J)h Fw(,)42 b(it)d(follo)m(ws)g(that)h Fu(\031)t Fw(\()p Fu(I)8 b(J)1262 5403 y Fp(0)1301 5439 y Fw(\))40 b(is)g(dense)i(in)d Fu(\031)t Fw(\()p Fu(I)8 b(J)h Fw(\).)66 b(On)40 b(the)h(other)f(hand,)j(the)d(subspace)94 5555 y Fu(\031)t Fw(\()p Fu(I)8 b(f)290 5570 y Fp(1)352 5555 y Fw(+)23 b Fr(\001)17 b(\001)g(\001)k Fw(+)i Fu(I)8 b(f)788 5570 y Fq(N)855 5555 y Fw(\))29 b Fr(\022)h Fw(T)-8 b(r\()p Fu(V)1245 5519 y Fq(?)1285 5555 y Fw(\))33 b(is)g(closed)h(b)m (y)g(Lemmas)2268 5555 y SDict begin H.S end 2268 5555 a Fw(13.7)2441 5491 y SDict begin H.R end 2441 5491 a 2441 5555 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.13.7) cvn H.B /ANN pdfmark end 2441 5555 a 34 w Fw(and)2665 5555 y SDict begin H.S end 2665 5555 a Fw(13.6)2839 5491 y SDict begin H.R end 2839 5491 a 2839 5555 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /BS <</S/S/W 1>> /Subtype /Link /Dest (theorem.13.6) cvn H.B /ANN pdfmark end 2839 5555 a Fw(.)46 b(W)-8 b(e)34 b(conclude)g(that)94 5671 y Fu(\031)t Fw(\()p Fu(I)8 b(J)305 5635 y Fp(0)345 5671 y Fw(\))27 b(=)h Fu(\031)t Fw(\()p Fu(I)8 b(f)710 5686 y Fp(1)771 5671 y Fw(+)22 b Fr(\001)17 b(\001)g(\001)k Fw(+)h Fu(I)8 b(f)1205 5686 y Fq(N)1272 5671 y Fw(\))27 b(=)h Fu(\031)t Fw(\()p Fu(I)8 b(J)h Fw(\),)32 b(as)h(required.)1339 b Fh(\003)p eop end end %%Page: 57 57 TeXDict begin HPSdict begin 57 56 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.57) cvn H.B /DEST pdfmark end 94 218 a 1179 w Fs(QUIVERS)31 b(WITH)i(POTENTIALS)g(I)1102 b(57)1411 417 y Fx(A)m(ckno)n(wledgments)94 475 y SDict begin H.S end 94 475 a 94 475 a SDict begin 14 H.A end 94 475 a 94 475 a SDict begin [ /View [/XYZ H.V] /Dest (section*.2) cvn H.B /DEST pdfmark end 94 475 a 194 591 a Fw(W)-8 b(e)54 b(thank)f(Victor)f(Ginzburg)h(for)f(helpful)g(commen)m(ts)h(and)h (bibliographic)49 b(guidance,)94 708 y(Daniel)c(Labardini)f(F)-8 b(ragoso)46 b(for)g(careful)g(reading)f(of)h(the)h(man)m(uscript)f(and) h(man)m(y)f(useful)94 824 y(suggestions)28 b(and)g(Christopher)g (Herzog)g(for)f(making)f(us)i(a)m(w)m(are)g(of)f(and)h(discussing)f (the)h(use)h(of)94 940 y(quiv)m(ers)k(with)f(sup)s(erp)s(oten)m(tials)f (in)g(theoretical)f(ph)m(ysics.)45 b(After)32 b(this)f(pap)s(er)h(w)m (as)h(completed,)94 1056 y(w)m(e)k(ha)m(v)m(e)f(b)s(een)g(informed)e(b) m(y)i(Maxim)e(Kon)m(tsevic)m(h)j(that)e(he)g(has)h(redisco)m(v)m(ered)h (some)e(of)g(the)94 1172 y(results)e(in)f(Sections)h(6)f(and)h(7)f(w)m (orking)h(in)f(the)h(con)m(text)h(of)e Fu(A)2432 1187 y Fn(1)2507 1172 y Fw(-categories.)1587 1454 y Fx(References)94 1512 y SDict begin H.S end 94 1512 a 94 1512 a SDict begin 14 H.A end 94 1512 a 94 1512 a SDict begin [ /View [/XYZ H.V] /Dest (section*.3) cvn H.B /DEST pdfmark end 94 1512 a 94 1512 a SDict begin H.S end 94 1512 a 94 1512 a SDict begin 12 H.A end 94 1512 a 94 1512 a SDict begin [ /View [/XYZ H.V] /Dest (cite.ca3) cvn H.B /DEST pdfmark end 94 1512 a 184 1611 a Fz([1])42 b(A.)31 b(Berenstein,)g(S.)h(F)-7 b(omin)31 b(and)g(A.)g(Zelevinsky)-7 b(,)32 b(Cluster)e(algebras)f(I)r (I)r(I:)j(Upp)r(er)f(b)r(ounds)h(and)f(double)314 1711 y(Bruhat)c(cells,)g Fc(Duk)n(e)h(Math.)f(J.)h Fb(126)f Fz(\(2005\),)f(1{52.)94 1732 y SDict begin H.S end 94 1732 a 94 1732 a SDict begin 12 H.A end 94 1732 a 94 1732 a SDict begin [ /View [/XYZ H.V] /Dest (cite.bd) cvn H.B /DEST pdfmark end 94 1732 a 184 1811 a Fz([2])42 b(D.)79 b(Berenstein)g(and)g(M.R.)g(Douglas,)91 b(Seib)r(erg)79 b(Dualit)n(y)g(for)f(Quiv)n(er)g(Gauge)h(Theories,)314 1910 y Fa(hep-th/0207027)p Fz(.)94 1929 y SDict begin H.S end 94 1929 a 94 1929 a SDict begin 12 H.A end 94 1929 a 94 1929 a SDict begin [ /View [/XYZ H.V] /Dest (cite.bgp) cvn H.B /DEST pdfmark end 94 1929 a 184 2010 a Fz([3])42 b(J.)26 b(Bernstein,)h(I.)g(Gelfand)h(and)f(V.)g(P)n(onomarev,)e(Co)n (xeter)g(functors)i(and)g(Gabriel's)f(theorem,)h Fc(Usp)r(ehi)314 2110 y(Mat.)g(Nauk)33 b Fb(28)27 b Fz(\(1973\),)f(no.)i(2\(170\),)e (19-33.)94 2130 y SDict begin H.S end 94 2130 a 94 2130 a SDict begin 12 H.A end 94 2130 a 94 2130 a SDict begin [ /View [/XYZ H.V] /Dest (cite.boc) cvn H.B /DEST pdfmark end 94 2130 a 184 2209 a Fz([4])42 b(R.)27 b(Bo)r(c)n(klandt,)g(Graded)g (Calabi)g(Y)-7 b(au)28 b(Algebras)e(of)i(dimension)f(3,)g Fa(math.RA/0603558)p Fz(.)94 2230 y SDict begin H.S end 94 2230 a 94 2230 a SDict begin 12 H.A end 94 2230 a 94 2230 a SDict begin [ /View [/XYZ H.V] /Dest (cite.blb) cvn H.B /DEST pdfmark end 94 2230 a 184 2309 a Fz([5])42 b(R.)24 b(Bo)r(c)n(klandt)f(and)h(L.)g(Le)f(Bruyn,)h(Nec)n(klace)f(Lie) h(algebras)e(and)i(noncomm)n(utativ)n(e)f(symplectic)h(geom-)314 2409 y(etry)-7 b(,)27 b Fc(Math.)h(Z.)f Fb(240)g Fz(\(2002\),)f(no.)i (1,)f(141{167.)94 2429 y SDict begin H.S end 94 2429 a 94 2429 a SDict begin 12 H.A end 94 2429 a 94 2429 a SDict begin [ /View [/XYZ H.V] /Dest (cite.b) cvn H.B /DEST pdfmark end 94 2429 a 184 2508 a Fz([6])42 b(V.)25 b(Braun,)g(On)g (Berenstein-Douglas-Seib)r(erg)e(Dualit)n(y)-7 b(,)26 b Fc(J.)f(High)g(Energy)f(Ph)n(ys.)g Fz(2003,)g(no.)i(1,)f(082,)f(21) 314 2608 y(pp.)94 2624 y SDict begin H.S end 94 2624 a 94 2624 a SDict begin 12 H.A end 94 2624 a 94 2624 a SDict begin [ /View [/XYZ H.V] /Dest (cite.BMRRT) cvn H.B /DEST pdfmark end 94 2624 a 184 2707 a Fz([7])42 b(A.)35 b(B.)g(Buan,)h(R.)g (Marsh,)g(M.)f(Reinek)n(e,)h(I.)f(Reiten)h(and)f(G.)g(T)-7 b(o)r(doro)n(v,)35 b(Tilting)g(theory)f(and)h(cluster)314 2807 y(com)n(binatorics,)25 b Fc(Adv.)j(Math.)g Fb(204)f Fz(\(2006\),)f(no.)i(2,)f(572-618.)94 2828 y SDict begin H.S end 94 2828 a 94 2828 a SDict begin 12 H.A end 94 2828 a 94 2828 a SDict begin [ /View [/XYZ H.V] /Dest (cite.BMR1) cvn H.B /DEST pdfmark end 94 2828 a 184 2907 a Fz([8])42 b(A.)33 b(B.)g(Buan,)h(R.)g(Marsh)e(and)h(I.)g(Reiten,)i (Cluster-tilted)f(algebras,)e Fc(T)-7 b(ran.)33 b(Amer.)g(Math.)g(So)r (c.)g Fb(359)314 3006 y Fz(\(2007\),)26 b(no.)h(1,)h(323-332.)94 3027 y SDict begin H.S end 94 3027 a 94 3027 a SDict begin 12 H.A end 94 3027 a 94 3027 a SDict begin [ /View [/XYZ H.V] /Dest (cite.BMR2) cvn H.B /DEST pdfmark end 94 3027 a 184 3106 a Fz([9])42 b(A.)31 b(B.)f(Buan,)i(R.)f(Marsh)f(and)g(I.)h (Reiten)g(Cluster-tilted)g(algebras)e(of)i(\014nite)g(represen)n (tation)e(t)n(yp)r(e,)j Fc(J.)314 3206 y(of)55 b(Algebra.)26 b Fb(306)h Fz(\(2006\),)g(no.)g(2,)g(412-431.)94 3226 y SDict begin H.S end 94 3226 a 94 3226 a SDict begin 12 H.A end 94 3226 a 94 3226 a SDict begin [ /View [/XYZ H.V] /Dest (cite.BMR3) cvn H.B /DEST pdfmark end 94 3226 a 143 3305 a Fz([10])41 b(A.)58 b(B.)f(Buan,)65 b(R.)58 b(Marsh)f(and)h(I.)g (Reiten,)65 b(Cluster)58 b(m)n(utation)f(via)h(quiv)n(er)e(represen)n (tations,)314 3405 y Fa(math.RT/0412077)o Fz(.)94 3412 y SDict begin H.S end 94 3412 a 94 3412 a SDict begin 12 H.A end 94 3412 a 94 3412 a SDict begin [ /View [/XYZ H.V] /Dest (cite.BR) cvn H.B /DEST pdfmark end 94 3412 a 143 3504 a Fz([11])41 b(C.R.)27 b(Butler)g(and)g(C.M.)g(Ringel,)g (Auslander-Reiten)g(sequences)f(with)i(few)f(middle)h(terms)f(and)g (appli-)314 3604 y(cations)f(to)i(string)f(algebras,)e Fc(Comm.)j(Algebra)f Fb(15)g Fz(\(1987\),)f(269{290.)94 3625 y SDict begin H.S end 94 3625 a 94 3625 a SDict begin 12 H.A end 94 3625 a 94 3625 a SDict begin [ /View [/XYZ H.V] /Dest (cite.CC) cvn H.B /DEST pdfmark end 94 3625 a 143 3704 a Fz([12])41 b(P)-7 b(.)37 b(Caldero)g(and)h(F.)h(Chap)r (oton,)h(Cluster)e(algebras)e(as)h(Hall)h(algebras)e(of)i(quiv)n(er)f (represen)n(tations,)314 3803 y Fc(Commen)n(t.)27 b(Math.)h(Helv.)g Fb(81)f Fz(\(2006\),)f(no.)i(3,)f(595-616.)94 3824 y SDict begin H.S end 94 3824 a 94 3824 a SDict begin 12 H.A end 94 3824 a 94 3824 a SDict begin [ /View [/XYZ H.V] /Dest (cite.CCS) cvn H.B /DEST pdfmark end 94 3824 a 143 3903 a Fz([13])41 b(P)-7 b(.)28 b(Caldero,)f(F.)i(Chap)r(oton)f(and)g (R.)h(Sc)n(hi\017er,)f(Quiv)n(ers)f(with)i(relations)e(and)h(cluster)g (tilted)h(algebras,)314 4003 y Fc(Algebras)d(and)55 b(Rep.)h(Theory)33 b Fb(9)28 b Fz(\(2006\),)e(359{376.)94 4023 y SDict begin H.S end 94 4023 a 94 4023 a SDict begin 12 H.A end 94 4023 a 94 4023 a SDict begin [ /View [/XYZ H.V] /Dest (cite.CK1) cvn H.B /DEST pdfmark end 94 4023 a 143 4102 a Fz([14])41 b(P)-7 b(.)21 b(Caldero)f(and)h(B.)g(Keller,)h(F)-7 b(rom)21 b(triangulated)f(categories)g(to)h(cluster)g(algebras,)g Fa(math.RT/0506018)o Fz(.)94 4123 y SDict begin H.S end 94 4123 a 94 4123 a SDict begin 12 H.A end 94 4123 a 94 4123 a SDict begin [ /View [/XYZ H.V] /Dest (cite.CK2) cvn H.B /DEST pdfmark end 94 4123 a 143 4202 a Fz([15])41 b(P)-7 b(.)76 b(Caldero)f(and)h(B.)g(Keller,)88 b(F)-7 b(rom)76 b(triangulated)f(categories)g(to)h(cluster)g(algebras)e(I)r (I,)314 4301 y Fa(math.RT/0510251)o Fz(.)94 4308 y SDict begin H.S end 94 4308 a 94 4308 a SDict begin 12 H.A end 94 4308 a 94 4308 a SDict begin [ /View [/XYZ H.V] /Dest (cite.calzel) cvn H.B /DEST pdfmark end 94 4308 a 143 4401 a Fz([16])41 b(P)-7 b(.)27 b(Caldero,)f(A.)i(Zelevinsky)-7 b(,)27 b(Lauren)n(t)f(expansions)h(in)g(cluster)g(algebras)f(via)h (quiv)n(er)f(represen)n(tations,)314 4501 y Fc(Mosco)n(w)g(Math.)h(J.)h Fb(6)f Fz(\(2006\),)g(no.)g(3,)g(411{429.)94 4521 y SDict begin H.S end 94 4521 a 94 4521 a SDict begin 12 H.A end 94 4521 a 94 4521 a SDict begin [ /View [/XYZ H.V] /Dest (cite.dm) cvn H.B /DEST pdfmark end 94 4521 a 143 4600 a Fz([17])41 b(M.R.)28 b(Douglas)e(and)i(G.)f(Mo)r(ore,)g(D-branes,)g (Quiv)n(ers)f(and)h(ALE)h(Instan)n(tons,)f Fa(hep-th/9603167)p Fz(.)94 4621 y SDict begin H.S end 94 4621 a 94 4621 a SDict begin 12 H.A end 94 4621 a 94 4621 a SDict begin [ /View [/XYZ H.V] /Dest (cite.ca1) cvn H.B /DEST pdfmark end 94 4621 a 143 4700 a Fz([18])41 b(S.)24 b(F)-7 b(omin)25 b(and)g(A.)g(Zelevinsky)-7 b(,)24 b(Cluster)h(algebras)d(I:)j(F)-7 b(oundations,)25 b Fc(J.)f(Amer.)h(Math.)g(So)r(c.)g Fb(15)f Fz(\(2002\),)314 4800 y(497{529.)94 4800 y SDict begin H.S end 94 4800 a 94 4800 a SDict begin 12 H.A end 94 4800 a 94 4800 a SDict begin [ /View [/XYZ H.V] /Dest (cite.ca2) cvn H.B /DEST pdfmark end 94 4800 a 143 4899 a Fz([19])41 b(S.)24 b(F)-7 b(omin)25 b(and)g(A.)g(Zelevinsky)-7 b(,)24 b(Cluster)g(algebras)f(I)r(I:)i(Finite)g(t)n(yp)r(e)g (classi\014cation,)e Fc(In)n(v)n(en)n(t.)h(Math.)h Fb(154)314 4999 y Fz(\(2003\),)h(63{121.)94 5020 y SDict begin H.S end 94 5020 a 94 5020 a SDict begin 12 H.A end 94 5020 a 94 5020 a SDict begin [ /View [/XYZ H.V] /Dest (cite.ca4) cvn H.B /DEST pdfmark end 94 5020 a 143 5098 a Fz([20])41 b(S.)d(F)-7 b(omin)39 b(and)f(A.)h(Zelevinsky)-7 b(,)40 b(Cluster)e(algebras)e(IV:)j(Co)r(e\016cien)n(ts,)h Fc(Comp.)f(Math.)f Fb(143)g Fz(\(2007\),)314 5198 y(112{164.)94 5198 y SDict begin H.S end 94 5198 a 94 5198 a SDict begin 12 H.A end 94 5198 a 94 5198 a SDict begin [ /View [/XYZ H.V] /Dest (cite.F) cvn H.B /DEST pdfmark end 94 5198 a 143 5298 a Fz([21])j(K.R.)19 b(F)-7 b(uller,)21 b(Biserial)e(Rings,)i Fc(Lecture)e(Notes)g(in)h(Mathematics)p Fz(,)h Fb(734)p Fz(,)f(Springer,)h(Berlin,)f(1979,)g(64{90.)94 5318 y SDict begin H.S end 94 5318 a 94 5318 a SDict begin 12 H.A end 94 5318 a 94 5318 a SDict begin [ /View [/XYZ H.V] /Dest (cite.gls) cvn H.B /DEST pdfmark end 94 5318 a 143 5397 a Fz([22])41 b(C.)26 b(Geiss,)g(B.)g(Leclerc)f(and)h(J.)g(Sc)n (hr\177)-42 b(oer,)25 b(Auslander)h(algebras)e(and)i(initial)g(seeds)g (for)g(cluster)f(algebras,)314 5497 y Fa(math.RT/0506405)o Fz(.)94 5504 y SDict begin H.S end 94 5504 a 94 5504 a SDict begin 12 H.A end 94 5504 a 94 5504 a SDict begin [ /View [/XYZ H.V] /Dest (cite.gkz) cvn H.B /DEST pdfmark end 94 5504 a 143 5597 a Fz([23])41 b(I.)24 b(Gelfand,)i(M.)e(Kaprano)n (v)e(and)i(A.)h(Zelevinsky)-7 b(,)24 b(Discriminan)n(ts,)h(Resultan)n (ts)f(and)g(Multidimensional)314 5696 y(Determinan)n(ts,)j(Birkh\177) -42 b(auser)26 b(Boston,)h(1994.)p eop end end %%Page: 58 58 TeXDict begin HPSdict begin 58 57 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 94 218 a SDict begin H.S end 94 218 a 94 218 a SDict begin H.R end 94 218 a 94 218 a SDict begin [ /View [/XYZ H.V] /Dest (page.58) cvn H.B /DEST pdfmark end 94 218 a Fs(58)492 b(HARM)25 b(DERKSEN,)f(JERZY)i(WEYMAN,)f(AND)f(ANDREI)g(ZELEVINSKY)94 334 y SDict begin H.S end 94 334 a 94 334 a SDict begin 12 H.A end 94 334 a 94 334 a SDict begin [ /View [/XYZ H.V] /Dest (cite.gi1) cvn H.B /DEST pdfmark end 94 334 a 143 417 a Fz([24])41 b(V.)35 b(Ginzburg,)i(Non-comm)n(utativ)n(e)c (symplectic)j(geometry)-7 b(,)35 b(quiv)n(er)f(v)-5 b(arieties,)37 b(and)e(op)r(erads,)g Fc(Math.)314 517 y(Res.)27 b(Lett.)h Fb(8)g Fz(\(2001\),)e(no.)h(3,)h(377{400.)94 537 y SDict begin H.S end 94 537 a 94 537 a SDict begin 12 H.A end 94 537 a 94 537 a SDict begin [ /View [/XYZ H.V] /Dest (cite.gi) cvn H.B /DEST pdfmark end 94 537 a 143 616 a Fz([25])41 b(V.)28 b(Ginzburg,)f(Calabi-Y)-7 b(au)26 b(algebras,)g Fa(math.AG/0612139)p Fz(.)94 637 y SDict begin H.S end 94 637 a 94 637 a SDict begin 12 H.A end 94 637 a 94 637 a SDict begin [ /View [/XYZ H.V] /Dest (cite.ir) cvn H.B /DEST pdfmark end 94 637 a 143 716 a Fz([26])41 b(O.)c(Iy)n(ama)f(and)h (I.)h(Reiten,)i(F)-7 b(omin-Zelevinsky)37 b(m)n(utation)g(and)g (tilting)h(mo)r(dules)g(o)n(v)n(er)e(Calabi-Y)-7 b(au)314 815 y(algebras,)25 b Fa(math.RT/0605136)p Fz(.)94 832 y SDict begin H.S end 94 832 a 94 832 a SDict begin 12 H.A end 94 832 a 94 832 a SDict begin [ /View [/XYZ H.V] /Dest (cite.kr) cvn H.B /DEST pdfmark end 94 832 a 143 915 a Fz([27])41 b(B.)27 b(Keller)g(and)g(I.)h(Reiten,)g(Acyclic)g (Calabi-Y)-7 b(au)26 b(categories,)g Fa(math.RT/0610594)p Fz(.)94 936 y SDict begin H.S end 94 936 a 94 936 a SDict begin 12 H.A end 94 936 a 94 936 a SDict begin [ /View [/XYZ H.V] /Dest (cite.mrz) cvn H.B /DEST pdfmark end 94 936 a 143 1015 a Fz([28])41 b(R.)20 b(Marsh,)h(M.)g(Reinek)n(e)f(and)h (A.)g(Zelevinsky)-7 b(,)21 b(Generalized)f(asso)r(ciahedra)e(via)i (quiv)n(er)f(represen)n(tations.)314 1114 y Fc(T)-7 b(rans.)26 b(Amer.)i(Math.)g(So)r(c.)f Fb(355)g Fz(\(2003\),)f(no.)i(10,)f (4171{4186.)94 1135 y SDict begin H.S end 94 1135 a 94 1135 a SDict begin 12 H.A end 94 1135 a 94 1135 a SDict begin [ /View [/XYZ H.V] /Dest (cite.rss) cvn H.B /DEST pdfmark end 94 1135 a 143 1214 a Fz([29])41 b(G.-C.)22 b(Rota,)h(B.)f(Sagan) f(and)h(P)-7 b(.)22 b(Stein,)i(A)e(cyclic)g(deriv)-5 b(ativ)n(e)21 b(in)i(noncomm)n(utativ)n(e)e(algebra.)g Fc(J.)h(Algebra)314 1314 y Fb(64)27 b Fz(\(1980\),)f(no.)h(1,)h(54{75.) 194 1500 y Fy(Dep)-6 b(ar)g(tment)32 b(of)f(Ma)-6 b(thema)g(tics,)32 b(University)e(of)h(Michigan,)f(Ann)g(Arbor,)g(MI)g(48109,)g(USA)194 1600 y Ft(E-mail)h(addr)l(ess)7 b Fz(:)38 b Fa(hderksen@umich.e)o(du) 194 1769 y Fy(Dep)-6 b(ar)g(tment)33 b(of)f(Ma)-6 b(thema)g(tics,)32 b(Nor)-6 b(theastern)32 b(University,)e(Boston,)h(MA)g(02115)194 1869 y Ft(E-mail)g(addr)l(ess)7 b Fz(:)38 b Fa(j.weyman@neu.edu)194 2038 y Fy(Dep)-6 b(ar)g(tment)33 b(of)f(Ma)-6 b(thema)g(tics,)32 b(Nor)-6 b(theastern)32 b(University,)e(Boston,)h(MA)g(02115)194 2138 y Ft(E-mail)g(addr)l(ess)7 b Fz(:)38 b Fa(andrei@neu.edu)p eop end end %%Trailer end userdict /end-hook known{end-hook}if %%EOF
Simpan