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transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll
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pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack}
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get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip
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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
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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
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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
%%BeginFont: CMTT8
%!PS-AdobeFont-1.1: CMTT8 1.0
%%CreationDate: 1991 Aug 20 16:46:05
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTT8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch true def
end readonly def
/FontName /CMTT8 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 /.notdef put
readonly def
/FontBBox{-5 -232 545 699}readonly def
/UniqueID 5000830 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMCSC10
%!PS-AdobeFont-1.1: CMCSC10 1.0
%%CreationDate: 1991 Aug 18 17:46:49
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMCSC10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMCSC10 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 /.notdef put
readonly def
/FontBBox{14 -250 1077 750}readonly def
/UniqueID 5000772 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMBX8
%!PS-AdobeFont-1.1: CMBX8 1.0
%%CreationDate: 1991 Aug 20 16:36:07
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMBX8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX8 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 /.notdef put
readonly def
/FontBBox{-59 -250 1235 750}readonly def
/UniqueID 5000766 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR5
%!PS-AdobeFont-1.1: CMR5 1.00B
%%CreationDate: 1992 Feb 19 19:55:02
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR5) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR5 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 /.notdef put
readonly def
/FontBBox{-341 -250 1304 965}readonly def
/UniqueID 5000788 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: MSAM10
%!PS-AdobeFont-1.1: MSAM10 2.1
%%CreationDate: 1993 Sep 17 09:05:00
% Math Symbol fonts were designed by the American Mathematical Society.
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (MSAM10) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /MSAM10 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 3 /square put
readonly def
/FontBBox{8 -463 1331 1003}readonly def
/UniqueID 5031981 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMTI10
%!PS-AdobeFont-1.1: CMTI10 1.00B
%%CreationDate: 1992 Feb 19 19:56:16
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTI10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMTI10 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 /.notdef put
readonly def
/FontBBox{-163 -250 1146 969}readonly def
/UniqueID 5000828 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI8
%!PS-AdobeFont-1.1: CMMI8 1.100
%%CreationDate: 1996 Jul 23 07:53:54
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI8 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 /.notdef put
readonly def
/FontBBox{-24 -250 1110 750}readonly def
/UniqueID 5087383 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSY5
%!PS-AdobeFont-1.1: CMSY5 1.0
%%CreationDate: 1991 Aug 15 07:21:16
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY5) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY5 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 /.notdef put
readonly def
/FontBBox{21 -944 1448 791}readonly def
/UniqueID 5000815 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI5
%!PS-AdobeFont-1.1: CMMI5 1.100
%%CreationDate: 1996 Aug 02 08:21:10
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI5) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI5 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 /.notdef put
readonly def
/FontBBox{37 -250 1349 750}readonly def
/UniqueID 5087380 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSY8
%!PS-AdobeFont-1.1: CMSY8 1.0
%%CreationDate: 1991 Aug 15 07:22:10
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY8 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
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readonly def
/FontBBox{-30 -955 1185 779}readonly def
/UniqueID 5000818 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMTI8
%!PS-AdobeFont-1.1: CMTI8 1.0
%%CreationDate: 1991 Aug 18 21:07:42
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTI8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMTI8 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 /.notdef put
readonly def
/FontBBox{-35 -250 1190 750}readonly def
/UniqueID 5000826 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR8
%!PS-AdobeFont-1.1: CMR8 1.0
%%CreationDate: 1991 Aug 20 16:39:40
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR8 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 /.notdef put
readonly def
/FontBBox{-36 -250 1070 750}readonly def
/UniqueID 5000791 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: EUFM10
%!PS-AdobeFont-1.1: EUFM10 2.1
%%CreationDate: 1992 Nov 20 17:36:20
% Euler fonts were designed by Hermann Zapf.
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (EUFM10) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /EUFM10 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 74 /J put
readonly def
/FontBBox{-26 -224 1055 741}readonly def
/UniqueID 5031986 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMEX10
%!PS-AdobeFont-1.1: CMEX10 1.00
%%CreationDate: 1992 Jul 23 21:22:48
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMEX10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMEX10 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 /parenleftbig put
dup 1 /parenrightbig put
dup 2 /bracketleftbig put
dup 3 /bracketrightbig put
dup 16 /parenleftBig put
dup 17 /parenrightBig put
dup 20 /bracketleftbigg put
dup 21 /bracketrightbigg put
dup 50 /bracketlefttp put
dup 51 /bracketrighttp put
dup 52 /bracketleftbt put
dup 53 /bracketrightbt put
dup 54 /bracketleftex put
dup 55 /bracketrightex put
dup 63 /arrowvertex put
dup 80 /summationtext put
dup 88 /summationdisplay put
dup 104 /bracketleftBig put
dup 105 /bracketrightBig put
dup 112 /radicalbig put
dup 113 /radicalBig put
dup 114 /radicalbigg put
dup 121 /arrowbt put
readonly def
/FontBBox{-24 -2960 1454 772}readonly def
/UniqueID 5000774 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: MSBM10
%!PS-AdobeFont-1.1: MSBM10 2.1
%%CreationDate: 1993 Sep 17 11:10:37
% Math Symbol fonts were designed by the American Mathematical Society.
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (MSBM10) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
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end readonly def
/FontName /MSBM10 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 67 /C put
readonly def
/FontBBox{-55 -420 2343 920}readonly def
/UniqueID 5031982 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSY7
%!PS-AdobeFont-1.1: CMSY7 1.0
%%CreationDate: 1991 Aug 15 07:21:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY7 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 /.notdef put
readonly def
/FontBBox{-15 -951 1252 782}readonly def
/UniqueID 5000817 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR7
%!PS-AdobeFont-1.1: CMR7 1.0
%%CreationDate: 1991 Aug 20 16:39:21
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR7 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 /.notdef put
readonly def
/FontBBox{-27 -250 1122 750}readonly def
/UniqueID 5000790 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI10
%!PS-AdobeFont-1.1: CMMI10 1.100
%%CreationDate: 1996 Jul 23 07:53:57
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI10 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 /.notdef put
readonly def
/FontBBox{-32 -250 1048 750}readonly def
/UniqueID 5087385 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI7
%!PS-AdobeFont-1.1: CMMI7 1.100
%%CreationDate: 1996 Jul 23 07:53:53
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI7 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 /.notdef put
readonly def
/FontBBox{0 -250 1171 750}readonly def
/UniqueID 5087382 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR10
%!PS-AdobeFont-1.1: CMR10 1.00B
%%CreationDate: 1992 Feb 19 19:54:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR10 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 /.notdef put
readonly def
/FontBBox{-251 -250 1009 969}readonly def
/UniqueID 5000793 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSY10
%!PS-AdobeFont-1.1: CMSY10 1.0
%%CreationDate: 1991 Aug 15 07:20:57
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY10 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 /.notdef put
readonly def
/FontBBox{-29 -960 1116 775}readonly def
/UniqueID 5000820 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMBX10
%!PS-AdobeFont-1.1: CMBX10 1.00B
%%CreationDate: 1992 Feb 19 19:54:06
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMBX10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX10 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 /.notdef put
readonly def
/FontBBox{-301 -250 1164 946}readonly def
/UniqueID 5000768 def
currentdict end
currentfile eexec
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showpage
%%Trailer
end
%%EOF

%%EndDocument
 @endspecial 0 1202 a Ft(with)33 b(the)g(relations)f
Fr(\015)733 1172 y Fp(\003)728 1224 y Fs(i)771 1202 y
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y Fs(i)1036 1202 y Ft(,)1093 1140 y Fn(P)1181 1161 y
Fs(n)1181 1227 y(i)p Fq(=1)1306 1202 y Fr(\013)1359 1214
y Fs(i)1387 1202 y Fr(\015)1430 1214 y Fs(i)1457 1202
y Fr(\015)1505 1172 y Fp(\003)1500 1224 y Fs(i)1575 1202
y Ft(=)g Fr(")1710 1214 y Fq(0)1780 1202 y Ft(\(for)i(the)g(quiv)n(ers)
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b(\013)1120 1302 y Ft(,)28 b(since)f(the)h Fv(\003)p
Ft(-algebra)d Fv(P)1935 1315 y Fs(n;)r(~)-36 b(\013)2071
1302 y Ft(can)27 b(b)r(e)h(obtained)f(from)g(the)0 1402
y Fv(\003)p Ft(-quiv)n(er)h(as)h(a)h(factor)f(of)h(the)g(algebra)e(of)i
(paths)g(with)g(resp)r(ect)g(to)f(the)i Fv(\003)p Ft(-ideal)e
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y Fs(n;)r(~)g(\013)1155 1601 y Ft(\))27 b(has)e(at)h(least)f(one)h
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b Ft(\(\005\))0 1700 y(in)j(the)g(category)e(of)h(separable)f(Hilb)r
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b Fv(Q)1541 1813 y Fs(n;)r(~)-36 b(\013)1649 1800 y Ft(.)100
1900 y(If)28 b Fr(\031)s Ft(\()p Fr(p)307 1912 y Fs(i)335
1900 y Ft(\))23 b(=)g Fr(P)531 1912 y Fs(i)559 1900 y
Ft(,)k(\005\()p Fr(\015)746 1912 y Fs(i)774 1900 y Ft(\))d(=)e(\000)969
1912 y Fs(i)997 1900 y Ft(,)28 b(then)g(w)n(e)f(ha)n(v)n(e)0
2112 y(\(3\))852 2062 y Fr(P)905 2074 y Fs(i)956 2062
y Ft(=)c Fr(P)1109 2032 y Fp(\003)1097 2084 y Fs(i)1170
2062 y Ft(=)f Fr(P)1322 2032 y Fq(2)1310 2084 y Fs(i)1360
2062 y Fr(;)1593 2000 y Fn(P)1681 2020 y Fs(n)1681 2087
y(i)p Fq(=1)1806 2062 y Fr(\013)1859 2074 y Fs(i)1887
2062 y Fr(P)1940 2074 y Fs(i)1991 2062 y Ft(=)h Fr(I)2115
2074 y Fq(0)2152 2062 y Ft(;)934 2162 y(\000)986 2132
y Fp(\003)986 2183 y Fs(i)1024 2162 y Ft(\000)1076 2174
y Fs(i)1126 2162 y Ft(=)g Fr(I)1250 2174 y Fs(i)1278
2162 y Fr(;)1549 2100 y Fn(P)1636 2120 y Fs(n)1636 2187
y(i)p Fq(=1)1762 2162 y Fr(\013)1815 2174 y Fs(i)1843
2162 y Ft(\000)1895 2174 y Fs(i)1922 2162 y Ft(\000)1974
2132 y Fp(\003)1974 2183 y Fs(i)2035 2162 y Ft(=)g Fr(I)2159
2174 y Fq(0)2197 2162 y Fr(:)312 2319 y Fu(2.)45 b(Equiv)-7
b(alence)30 b(of)g(the)h(ca)-6 b(tegories)32 b Fr(R)q(ep)e
Fv(P)2004 2332 y Fs(n;)r(~)-36 b(\013)2142 2319 y Fu(and)31
b Fr(R)q(ep)e Fv(Q)2569 2332 y Fs(n;)r(~)-36 b(\013)100
2468 y Ft(Let)40 b(us)g(no)n(w)f(construct)h(a)f(functor)h
Fr(F)56 b Ft(:)84 b Fr(R)q(ep)39 b Fv(P)1783 2481 y Fs(n;)r(~)-36
b(\013)1935 2468 y Fv(\000)-14 b(!)44 b Fr(R)q(ep)39
b Fv(Q)2365 2481 y Fs(n;)r(~)-36 b(\013)2473 2468 y Ft(.)74
b(Let)40 b Fr(\031)k Ft(b)r(e)c(a)0 2568 y(represen)n(tation)33
b(of)h(the)h(quiv)n(er)e Fr(R)q(ep)g Fv(P)1295 2581 y
Fs(n;)r(~)-36 b(\013)1403 2568 y Ft(,)36 b Fr(\031)s
Ft(\()p Fr(p)1586 2580 y Fs(i)1614 2568 y Ft(\))f(=)f
Fr(P)1833 2580 y Fs(i)1895 2568 y Ft(orthogonal)e(pro)5
b(jections)33 b(on)h(the)0 2667 y(space)27 b Fr(H)291
2679 y Fq(0)328 2667 y Ft(,)h Fr(H)448 2679 y Fs(i)499
2667 y Ft(=)23 b Fr(I)7 b(m)28 b(P)784 2679 y Fs(i)812
2667 y Ft(,)g(and)f(\000)1076 2679 y Fs(i)1132 2667 y
Ft(the)h(natural)f(em)n(b)r(eddings)g(\(isometries\))h(of)f
Fr(H)2635 2679 y Fs(i)2691 2667 y Ft(in)n(to)g Fr(H)2928
2679 y Fq(0)2966 2667 y Ft(.)0 2767 y(Then)34 b(\000)275
2737 y Fp(\003)275 2789 y Fs(i)347 2767 y Ft(:)g Fr(H)473
2779 y Fq(0)544 2767 y Fv(\000)-14 b(!)34 b Fr(H)781
2779 y Fs(i)843 2767 y Ft(are)f(epimorphisms)g(and)h(\000)1741
2737 y Fp(\003)1741 2789 y Fs(i)1779 2767 y Ft(\000)1831
2779 y Fs(i)1892 2767 y Ft(=)g Fr(I)2027 2779 y Fs(i)2055
2767 y Ft(.)56 b(De\014ne)35 b(a)f(represen)n(tation)0
2867 y Fr(F)12 b Ft(\()p Fr(\031)s Ft(\))24 b(=)e(\005)28
b(of)g(the)g(quiv)n(er)f Fv(Q)938 2880 y Fs(n;)r(~)-36
b(\013)1073 2867 y Ft(as)27 b(follo)n(ws:)36 b(\005\()p
Fr(i)p Ft(\))23 b(=)g Fr(H)1815 2879 y Fs(i)1843 2867
y Ft(,)k(\005\()p Fr(\015)2030 2879 y Fs(i)2059 2867
y Ft(\))c(=)g(\000)2254 2879 y Fs(i)2281 2867 y Ft(,)28
b(\005\()p Fr(\015)2474 2836 y Fp(\003)2469 2888 y Fs(i)2512
2867 y Ft(\))c(=)e(\000)2707 2836 y Fp(\003)2707 2888
y Fs(i)2745 2867 y Ft(.)100 2966 y(Let)39 b Fr(C)319
2978 y Fq(0)396 2966 y Ft(b)r(e)g(a)g(morphism)g(of)g(the)h(represen)n
(tation)d Fr(\031)43 b Ft(in)n(to)c(a)g(represen)n(tation)i(~)-46
b Fr(\031)s Ft(,)43 b(that)0 3067 y(is,)c Fr(C)177 3079
y Fq(0)214 3067 y Fr(P)267 3079 y Fs(i)333 3067 y Ft(=)454
3046 y(~)436 3067 y Fr(P)489 3079 y Fs(i)517 3067 y Fr(C)576
3079 y Fq(0)613 3067 y Ft(.)64 b(Denote)37 b(b)n(y)f
Fr(C)1177 3079 y Fs(i)1242 3067 y Ft(the)g(restriction)g(of)g(the)h(op)
r(erator)e Fr(C)2458 3079 y Fq(0)2532 3067 y Ft(to)i(the)g(space)0
3167 y Fr(H)69 3179 y Fs(i)125 3167 y Ft(=)28 b Fr(I)7
b(m)31 b(P)418 3179 y Fs(i)446 3167 y Ft(.)47 b(Then)31
b(\()p Fr(C)827 3179 y Fq(0)865 3167 y Fr(;)14 b(C)961
3179 y Fq(1)998 3167 y Fr(;)g(C)1094 3179 y Fq(2)1132
3167 y Fr(;)g(:)g(:)g(:)f(;)h(C)1375 3179 y Fs(n)1421
3167 y Ft(\))31 b(is)g(a)f(morphism)h(from)f(the)h(represen)n(tation)f
(\005)0 3272 y(in)n(to)j(the)i(represen)n(tation)883
3251 y(~)873 3272 y(\005.)55 b(Set)34 b Fr(F)12 b Ft(\()p
Fr(C)1318 3284 y Fq(0)1356 3272 y Ft(\))34 b(=)e(\()p
Fr(C)1610 3284 y Fq(0)1648 3272 y Fr(;)14 b(C)1744 3284
y Fq(1)1782 3272 y Fr(;)g(C)1878 3284 y Fq(2)1915 3272
y Fr(;)g(:)g(:)g(:)g(;)g(C)2159 3284 y Fs(n)2204 3272
y Ft(\).)56 b(F)-7 b(rom)33 b(relations)g(\(2\))0 3372
y(and)27 b(\(3\),)h(it)g(is)g(easy)e(to)i(get)f(that)0
3686 y(\(4\))1072 3548 y(1\))83 b Fr(C)1288 3560 y Fs(i)1339
3548 y Ft(=)1432 3527 y(~)1427 3548 y(\000)1479 3514
y Fp(\003)1479 3569 y Fs(i)1517 3548 y Fr(C)1576 3560
y Fq(0)1613 3548 y Ft(\000)1665 3560 y Fs(i)1693 3548
y Fr(;)1072 3752 y Ft(2\))g Fr(C)1288 3764 y Fq(0)1349
3752 y Ft(=)1476 3648 y Fs(n)1436 3673 y Fn(X)1442 3850
y Fs(i)p Fq(=1)1570 3752 y Fr(\013)1623 3764 y Fs(i)1656
3731 y Ft(~)1651 3752 y(\000)1703 3764 y Fs(i)1730 3752
y Fr(C)1789 3764 y Fs(i)1817 3752 y Ft(\000)1869 3718
y Fp(\003)1869 3772 y Fs(i)1907 3752 y Fr(:)0 4011 y
Ft(One)36 b(can)h(also)e(construct)h(an)h(\\in)n(v)n(erse")d(functor,)
39 b Fr(F)1775 3981 y Fq(\()p Fp(\000)p Fq(1\))1954 4011
y Ft(:)f Fr(R)q(ep)e Fv(Q)2264 4024 y Fs(n;)r(~)-36 b(\013)2410
4011 y Fv(\000)-14 b(!)75 b Fr(R)q(ep)36 b Fv(P)2858
4024 y Fs(n;)r(~)-36 b(\013)2966 4011 y Ft(,)0 4118 y
Fr(F)65 4087 y Fq(\()p Fp(\000)p Fq(1\))206 4118 y Ft(\(\005\))27
b(=)e Fr(\031)s Ft(,)30 b(where)f Fr(\031)s Ft(\()p Fr(p)918
4130 y Fs(i)946 4118 y Ft(\))d(=)g(\000)1147 4130 y Fs(i)1175
4118 y Ft(\000)1227 4087 y Fp(\003)1227 4139 y Fs(i)1265
4118 y Ft(,)j(and)h(if)f(\()p Fr(C)1649 4130 y Fq(0)1687
4118 y Fr(;)14 b(C)1783 4130 y Fq(1)1821 4118 y Fr(;)g(C)1917
4130 y Fq(2)1954 4118 y Fr(;)g(:)g(:)g(:)g(;)g(C)2198
4130 y Fs(n)2243 4118 y Ft(\))30 b(is)f(a)g(morphism)g(of)g(a)0
4224 y(represen)n(tation)c(\005)i(in)n(to)f(a)g(represen)n(tation)1418
4203 y(~)1408 4224 y(\005,)h(then)g Fr(F)1773 4194 y
Fq(\()p Fp(\000)p Fq(1\))1914 4224 y Ft(\()p Fr(C)2005
4236 y Fq(0)2043 4224 y Fr(;)14 b(C)2139 4236 y Fq(1)2176
4224 y Fr(;)g(C)2272 4236 y Fq(2)2310 4224 y Fr(;)g(:)g(:)g(:)f(;)h(C)
2553 4236 y Fs(n)2599 4224 y Ft(\))23 b(=)g Fr(C)2801
4236 y Fq(0)2865 4224 y Ft(is)j(a)0 4323 y(morphism)h(from)g(the)h
(represen)n(tation)e Fr(\031)31 b Ft(in)n(to)d(the)g(represen)n(tation)
i(~)-47 b Fr(\031)t Ft(.)0 4467 y Fw(Prop)s(osition)24
b(1.)47 b Ff(The)25 b(functors)f Fr(F)36 b Ff(and)25
b Fr(F)1424 4436 y Fq(\()p Fp(\000)p Fq(1\))1590 4467
y Ff(pr)l(ovide)h(an)e(e)l(quivalenc)l(e)i(of)f(the)g(c)l(ate)l(gories)
0 4566 y Fr(R)q(ep)k Fv(P)232 4579 y Fs(n;)r(~)-36 b(\013)369
4566 y Ff(and)31 b Fr(R)q(ep)d Fv(Q)772 4579 y Fs(n;)r(~)-36
b(\013)880 4566 y Ff(.)0 4709 y(Pr)l(o)l(of.)44 b Ft(The)28
b(pro)r(of)f(is)g(almost)g(clear.)36 b(It)28 b(follo)n(ws)f(from)g
(relation)g(\(4\))h(that)g(the)g(functor)f Fr(F)40 b
Ft(is)0 4809 y(strict)26 b(and)h(faithful.)37 b(An)n(y)27
b(represen)n(tation)e(\005)h(of)h(the)g(quiv)n(er)e Fv(Q)2060
4822 y Fs(n;)r(~)-36 b(\013)2195 4809 y Ft(is)26 b(equiv)-5
b(alen)n(t)26 b(to)h(one)f(of)0 4915 y(the)34 b(represen)n(tations)e
Fr(F)12 b Ft(\()p Fr(\031)s Ft(\),)36 b(for)d(example,)i(the)f
(represen)n(tation)e Fr(F)12 b(F)2286 4885 y Fq(\()p
Fp(\000)p Fq(1\))2426 4915 y Ft(\(\005\).)56 b(This)34
b(ends)0 5015 y(the)28 b(pro)r(of.)2568 b Fe(\003)p eop
end
%%Page: 4 4
TeXDict begin 4 3 bop 0 42 a Fl(4)1145 b(S.)30 b(A.)g(KR)n(UGL)-6
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Fr(S)k Fu(and)c Fr(T)41 b Fu(and)31 b(the)f(Co)n(xeter)i(functors)100
390 y Ft(Let)39 b(the)g(spaces)f Fr(H)46 b Ft(and)989
369 y(^)967 390 y Fr(H)g Ft(b)r(e)39 b(decomp)r(osed)g(in)n(to)f
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b Ft(=)22 b Fr(H)262 463 y Fq(\(1\))363 493 y Fv(\010)12
b Fr(H)516 463 y Fq(\(2\))616 493 y Fv(\010)g(\001)i(\001)g(\001)c
(\010)i Fr(H)953 463 y Fq(\()p Fs(n)p Fq(\))1049 493
y Ft(,)1119 472 y(^)1097 493 y Fr(H)30 b Ft(=)1305 472
y(^)1284 493 y Fr(H)1360 463 y Fq(\(1\))1460 493 y Fv(\010)1558
472 y Ft(^)1537 493 y Fr(H)1613 463 y Fq(\(2\))1713 493
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493 y Fr(H)2050 463 y Fq(\()p Fs(m)p Fq(\))2164 493 y
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431 y Fn(P)2655 452 y Fs(m)2655 518 y(i)p Fq(=1)2780
493 y Fv(\010P)2910 463 y Fq(\()p Fs(i)p Fq(\))0 605
y Ft(and)20 b Fr(I)207 617 y Fq(^)190 632 y Fs(H)276
605 y Ft(=)364 543 y Fn(P)452 563 y Fs(m)452 630 y(j)s
Fq(=1)585 605 y Fv(\010)668 584 y Ft(^)650 605 y Fv(P)715
575 y Fq(\()p Fs(j)s Fq(\))821 605 y Ft(b)r(e)g(the)h(corresp)r(onding)
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728 y(in)n(to)33 b(an)h(orthogonal)d(sum)j(of)f(orthogonal)f(pro)5
b(jections)32 b(and)h Fr(A)h Ft(:)66 b Fr(H)40 b Fv(\000)-14
b(!)2504 707 y Ft(^)2482 728 y Fr(H)40 b Ft(b)r(e)34
b(a)g(linear)0 831 y(op)r(erator.)69 b(Then,)42 b(in)n(tro)r(ducing)c
(the)h(notations)f Fr(A)1701 843 y Fs(ij)1802 831 y Ft(=)1927
810 y(^)1908 831 y Fv(P)1973 801 y Fq(\()p Fs(i)p Fq(\))2052
831 y Fr(A)p Fv(P)2179 801 y Fq(\()p Fs(j)s Fq(\))2266
831 y Ft(,)k(w)n(e)c(get)h(the)g(P)n(eirce)0 931 y(decomp)r(osition)27
b(of)h(the)g(op)r(erator)d Fr(A)p Ft(,)j(writing)g Fr(A)g
Ft(as)e(an)i(op)r(erator)e(matrix,)899 1253 y Fr(A)d
Ft(=)1072 1061 y Fn(2)1072 1207 y(6)1072 1260 y(4)1154
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Fr(:)p eop end
%%Page: 5 5
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925 y(^)0 946 y Fr(C)59 958 y Fq(0)97 946 y Fv(4)171
958 y Fs(k)234 946 y Ft(=)338 924 y(~)322 946 y Fv(4)396
958 y Fs(k)437 946 y Fr(C)496 958 y Fs(k)537 946 y Ft(,)e
Fr(C)646 958 y Fs(k)688 946 y Fv(4)762 916 y Fp(\003)762
970 y Fs(k)825 946 y Ft(=)929 924 y(~)913 946 y Fv(4)987
916 y Fp(\003)987 970 y Fs(k)1046 925 y Ft(^)1028 946
y Fr(C)1087 958 y Fq(0)1124 946 y Ft(,)h Fr(k)e Ft(=)d(1)p
Fr(;)14 b(:)g(:)g(:)f(;)h(n)p Ft(.)100 1048 y(W)-7 b(e)36
b(giv)n(e)f(a)h(detailed)g(pro)r(of)g(only)g(for)f(the)i(\014rst)f
(equalit)n(y;)k(the)c(second)g(one)f(is)h(pro)n(v)n(ed)0
1151 y(similarly)-7 b(.)35 b(W)-7 b(e)23 b(ha)n(v)n(e)707
1130 y(^)689 1151 y Fr(C)748 1163 y Fq(0)785 1151 y Fv(4)859
1163 y Fs(k)923 1151 y Ft(=)1011 1089 y Fn(P)1098 1110
y Fs(n)1098 1176 y(i)p Fq(=1)1210 1151 y Ft(\(1)9 b Fv(\000)g
Fr(\013)1420 1163 y Fs(i)1447 1151 y Ft(\))1496 1129
y(~)1479 1151 y Fv(4)1553 1163 y Fs(i)1581 1151 y Fr(C)1640
1163 y Fs(i)1668 1151 y Ft(\()p Fv(4)1774 1121 y Fp(\003)1774
1173 y Fs(i)1812 1151 y Fv(4)1886 1163 y Fs(k)1927 1151
y Ft(\).)36 b(In)23 b(virtue)g(of)g(\(5.1\))f(and)h(\(5.2\),)19
1267 y(^)0 1288 y Fr(C)59 1300 y Fq(0)97 1288 y Fv(4)171
1300 y Fs(k)234 1288 y Ft(=)g Fv(\000)401 1225 y Fn(P)488
1246 y Fs(n)488 1312 y(i)p Fq(=1)p Fs(;i)p Fp(6)p Fq(=)p
Fs(k)744 1191 y Fn(q)p 827 1191 370 4 v 837 1245 a Fs(\013)880
1253 y Fi(i)907 1245 y Fs(\013)950 1253 y Fi(j)981 1245
y Fq(\(1)p Fp(\000)p Fs(\013)1135 1253 y Fi(i)1161 1245
y Fq(\))p 837 1268 350 4 v 904 1316 a(\(1)p Fp(\000)p
Fs(\013)1058 1325 y Fi(k)1095 1316 y Fq(\))1213 1266
y Ft(~)1197 1288 y Fv(4)1271 1300 y Fs(i)1299 1288 y
Ft(\()p Fr(C)1390 1300 y Fs(i)1418 1288 y Ft(\000)1470
1257 y Fp(\003)1470 1309 y Fs(i)1508 1288 y Ft(\)\000)1592
1300 y Fs(k)1642 1288 y Ft(+)9 b(\(1)g Fv(\000)g Fr(\013)1926
1300 y Fs(k)1967 1288 y Ft(\))2016 1266 y(~)1999 1288
y Fv(4)2073 1300 y Fs(k)2114 1288 y Fr(C)2173 1300 y
Fs(k)2215 1288 y Ft(.)35 b(It)23 b(follo)n(ws)f(from)h(\(2.2\))0
1449 y(that)195 1428 y(^)177 1449 y Fr(C)236 1461 y Fq(0)273
1449 y Fv(4)347 1461 y Fs(k)411 1449 y Ft(=)g Fv(\000)578
1386 y Fn(P)665 1407 y Fs(n)665 1474 y(i)p Fq(=1)p Fs(;i)p
Fp(6)p Fq(=)p Fs(k)921 1352 y Fn(q)p 1004 1352 370 4
v 1014 1407 a Fs(\013)1057 1415 y Fi(i)1083 1407 y Fs(\013)1126
1415 y Fi(j)1158 1407 y Fq(\(1)p Fp(\000)p Fs(\013)1312
1415 y Fi(i)1338 1407 y Fq(\))p 1014 1430 350 4 v 1081
1477 a(\(1)p Fp(\000)p Fs(\013)1235 1486 y Fi(k)1271
1477 y Fq(\))1390 1427 y Ft(~)1374 1449 y Fv(4)1448 1461
y Fs(i)1481 1428 y Ft(~)1475 1449 y(\000)1527 1419 y
Fp(\003)1527 1470 y Fs(i)1565 1449 y Ft(\()p Fr(C)1656
1461 y Fq(0)1694 1449 y Ft(\000)1746 1461 y Fs(k)1787
1449 y Ft(\))12 b(+)g(\(1)g Fv(\000)g Fr(\013)2124 1461
y Fs(k)2164 1449 y Ft(\))2213 1427 y(~)2196 1449 y Fv(4)2270
1461 y Fs(k)2311 1449 y Fr(C)2370 1461 y Fs(k)2412 1449
y Ft(,)25 b(and)f(\(2.1\))g(giv)n(es)19 1589 y(^)0 1610
y Fr(C)59 1622 y Fq(0)97 1610 y Fv(4)171 1622 y Fs(k)234
1610 y Ft(=)f Fv(\000)401 1547 y Fn(P)488 1568 y Fs(n)488
1635 y(i)p Fq(=1)p Fs(;i)p Fp(6)p Fq(=)p Fs(k)744 1513
y Fn(q)p 827 1513 370 4 v 837 1568 a Fs(\013)880 1576
y Fi(i)907 1568 y Fs(\013)950 1576 y Fi(j)981 1568 y
Fq(\(1)p Fp(\000)p Fs(\013)1135 1576 y Fi(i)1161 1568
y Fq(\))p 837 1591 350 4 v 904 1638 a(\(1)p Fp(\000)p
Fs(\013)1058 1647 y Fi(k)1095 1638 y Fq(\))1213 1588
y Ft(~)1197 1610 y Fv(4)1271 1622 y Fs(i)1304 1589 y
Ft(~)1299 1610 y(\000)1351 1580 y Fp(\003)1351 1631 y
Fs(i)1394 1589 y Ft(~)1389 1610 y(\000)1441 1622 y Fs(k)1482
1610 y Fr(C)1541 1622 y Fs(k)1596 1610 y Ft(+)15 b(\(1)f
Fv(\000)h Fr(\013)1897 1622 y Fs(k)1938 1610 y Ft(\))1986
1588 y(~)1970 1610 y Fv(4)2044 1622 y Fs(k)2084 1610
y Fr(C)2143 1622 y Fs(k)2185 1610 y Ft(.)36 b(Since)26
b(\(5.4\))f(holds,)h(w)n(e)0 1746 y(ha)n(v)n(e)k(that)398
1725 y(^)379 1746 y Fr(C)438 1758 y Fq(0)475 1746 y Fv(4)549
1758 y Fs(k)619 1746 y Ft(=)f Fr(\013)766 1758 y Fs(k)823
1724 y Ft(~)807 1746 y Fv(4)881 1758 y Fs(k)927 1725
y Ft(~)922 1746 y(\000)974 1716 y Fp(\003)974 1770 y
Fs(k)1020 1725 y Ft(~)1014 1746 y(\000)1066 1758 y Fs(k)1107
1746 y Fr(C)1166 1758 y Fs(k)1228 1746 y Ft(+)21 b(\(1)g
Fv(\000)f Fr(\013)1547 1758 y Fs(k)1588 1746 y Ft(\))1637
1724 y(~)1620 1746 y Fv(4)1694 1758 y Fs(k)1735 1746
y Fr(C)1794 1758 y Fs(k)1864 1746 y Ft(=)29 b Fr(\013)2011
1758 y Fs(k)2068 1724 y Ft(~)2052 1746 y Fv(4)2126 1758
y Fs(k)2167 1746 y Fr(C)2226 1758 y Fs(k)2288 1746 y
Ft(+)21 b(\(1)f Fv(\000)h Fr(\013)2607 1758 y Fs(k)2648
1746 y Ft(\))2696 1724 y(~)2680 1746 y Fv(4)2754 1758
y Fs(k)2795 1746 y Fr(C)2854 1758 y Fs(k)2924 1746 y
Ft(=)16 1834 y(~)0 1856 y Fv(4)74 1868 y Fs(k)115 1856
y Fr(C)174 1868 y Fs(k)215 1856 y Ft(,)28 b(whic)n(h)f(w)n(as)g(to)g(b)
r(e)h(pro)n(v)n(ed.)100 1961 y(W)-7 b(e)28 b(th)n(us)g(ha)n(v)n(e)e
(constructed)i(a)f(functor)h Fr(G)c Ft(:)51 b Fr(R)q(ep)27
b Fv(Q)1827 1974 y Fs(n;)r(~)-36 b(\013)1959 1961 y Fv(\000)-15
b(!)24 b Fr(R)q(ep)j Fv(Q)2356 1987 y Fs(n;)2412 1975
y(~)2417 1987 y Fq(1)p Fp(\000)r Fs(~)-36 b(\013)2549
1961 y Ft(,)28 b Fr(G)p Ft(\(\005\))c(=)2914 1940 y(^)2903
1961 y(\005)q(,)0 2085 y Fr(G)p Ft(\()p Fr(C)156 2097
y Fq(0)194 2085 y Fr(;)14 b(C)290 2097 y Fq(1)328 2085
y Fr(;)g(:)g(:)g(:)f(;)h(C)571 2097 y Fs(n)617 2085 y
Ft(\))23 b(=)g(\()811 2064 y(^)792 2085 y Fr(C)851 2097
y Fq(0)888 2085 y Fr(;)14 b(C)984 2097 y Fq(1)1022 2085
y Fr(;)g(:)g(:)g(:)g(;)g(C)1266 2097 y Fs(n)1311 2085
y Ft(\).)100 2187 y(The)27 b(follo)n(wing)g(prop)r(osition,)f(lik)n(e)i
(Prop)r(osition)e(1,)h(is)g(also)g(immediate.)0 2343
y Fw(Prop)s(osition)33 b(2.)47 b Ff(The)33 b(functor)f
Fr(G)c Ft(:)59 b Fr(R)q(ep)31 b Fv(Q)1524 2356 y Fs(n;)r(~)-36
b(\013)1659 2343 y Fv(\000)-14 b(!)27 b Fr(R)q(ep)k Fv(Q)2064
2369 y Fs(n;)2120 2357 y(~)2125 2369 y Fq(1)p Fp(\000)r
Fs(~)-36 b(\013)2289 2343 y Ff(pr)l(ovides)34 b(an)e(e)l(quiva-)0
2446 y(lenc)l(e)e(b)l(etwe)l(en)f(the)h(c)l(ate)l(gories.)100
2602 y Ft(In)19 b(the)h(same)f(w)n(a)n(y)g(as)g(the)h(functor)f
Fr(F)1265 2572 y Fq(\()p Fp(\000)p Fq(1\))1426 2602 y
Ft(p)r(ermitted)h(to)g(go)e(from)i(the)g(category)d Fr(R)q(ep)i
Fv(Q)2881 2615 y Fs(n;)r(~)-36 b(\013)0 2714 y Ft(to)24
b(the)g(equiv)-5 b(alen)n(t)23 b(category)f Fr(R)q(ep)h
Fv(P)1184 2727 y Fs(n;)r(~)-36 b(\013)1292 2714 y Ft(,)25
b(using)e(a)h(similar)f(functor)2191 2693 y(^)2172 2714
y Fr(F)2237 2684 y Fq(\()p Fp(\000)p Fq(1\))2402 2714
y Ft(w)n(e)h(can)f(go)g(to)g(the)0 2825 y(category)16
b Fr(R)q(ep)i Fv(P)546 2851 y Fs(n;)602 2839 y(~)607
2851 y Fq(1)p Fp(\000)r Fs(~)-36 b(\013)762 2825 y Ft(:)822
2804 y(^)803 2825 y Fr(F)868 2795 y Fq(\()p Fp(\000)p
Fq(1\))1009 2825 y Ft(\()1051 2804 y(^)1041 2825 y(\005)q(\))23
b(=)k(^)-46 b Fr(\031)s Ft(,)20 b(where)j(^)-47 b Fr(\031)t
Ft(\()p Fr(p)1696 2837 y Fs(i)1723 2825 y Ft(\))24 b(=)e
Fv(4)1940 2837 y Fs(i)1968 2825 y Fv(4)2042 2795 y Fp(\003)2042
2847 y Fs(i)2098 2825 y Ft(and,)e(if)f(\()2393 2804 y(^)2374
2825 y Fr(C)2433 2837 y Fq(0)2471 2825 y Fr(;)14 b(C)2567
2837 y Fq(1)2604 2825 y Fr(;)g(C)2700 2837 y Fq(2)2738
2825 y Fr(;)g(:)g(:)g(:)g(;)g(C)2982 2837 y Fs(n)3027
2825 y Ft(\))p 3059 2862 42 116 v 0 2969 a(is)k(a)g(morphism)g(from)g
(the)h(represen)n(tation)1378 2948 y(^)1368 2969 y(\005)f(in)n(to)g
(the)h(represen)n(tation)2285 2928 y(^)2285 2948 y(~)2275
2969 y(\005,)h(then)2579 2948 y(^)2560 2969 y Fr(F)2625
2939 y Fq(\()p Fp(\000)p Fq(1\))2766 2969 y Ft(\()2817
2948 y(^)2798 2969 y Fr(C)2857 2981 y Fq(0)2895 2969
y Fr(;)14 b(C)2991 2981 y Fq(1)3029 2969 y Fr(;)g(C)3125
2981 y Fq(2)3162 2969 y Fr(;)g(:)g(:)g(:)g(;)g(C)3406
2981 y Fs(n)3451 2969 y Ft(\))23 b(=)p 3571 2989 42 120
v 19 3055 a(^)0 3076 y Fr(C)59 3088 y Fq(0)124 3076 y
Ft(is)k(a)g(morphism)g(from)g(the)h(represen)n(tation)i(^)-47
b Fr(\031)31 b Ft(in)n(to)c(the)h(represen)n(tation)2481
3057 y(^)2481 3076 y(~)-47 b Fr(\031)t Ft(.)37 b(De\014ne)28
b(no)n(w)0 3176 y(a)37 b(functor)f Fr(S)44 b Ft(:)75
b Fr(R)q(ep)37 b Fv(P)809 3189 y Fs(n;)r(~)-36 b(\013)955
3176 y Fv(\000)-14 b(!)38 b Fr(R)q(ep)f Fv(P)1367 3202
y Fs(n;)1423 3190 y(~)1428 3202 y Fq(1)o Fp(\000)r Fs(~)-36
b(\013)1559 3176 y Ft(,)40 b(0)e Fr(<)g(\013)1858 3188
y Fs(i)1924 3176 y Fr(<)h Ft(1,)g Fr(i)f Ft(=)g(1)p Fr(;)14
b(:)g(:)g(:)f(;)h(n)p Ft(,)39 b(to)e(b)r(e)g(the)0 3300
y(comp)r(osition)23 b(of)g(functors,)h Fr(S)k Ft(=)1076
3279 y(^)1057 3300 y Fr(F)1122 3270 y Fq(\()p Fp(\000)p
Fq(1\))1263 3300 y Fr(GF)12 b Ft(.)36 b(Since)24 b(eac)n(h)e(factor)h
(is)g(an)g(equiv)-5 b(alence)23 b(functor,)0 3399 y(w)n(e)k(get)h(the)g
(follo)n(wing)e(prop)r(osition.)0 3556 y Fw(Prop)s(osition)k(3.)47
b Ff(The)31 b(functor)f Fr(S)e Ft(:)54 b Fr(R)q(ep)29
b Fv(P)1486 3569 y Fs(n;)r(~)-36 b(\013)1617 3556 y Fv(\000)-14
b(!)23 b Fr(R)q(ep)29 b Fv(P)2006 3581 y Fs(n;)2062 3569
y(~)2067 3581 y Fq(1)p Fp(\000)r Fs(~)-36 b(\013)2199
3556 y Ff(,)30 b(de\014ne)l(d)g(for)h Ft(0)23 b Fr(<)g(\013)2873
3568 y Fs(i)2924 3556 y Fr(<)0 3659 y Ft(1)p Ff(,)30
b Fr(i)22 b Ft(=)h(1)p Fr(;)14 b(:)g(:)g(:)f(;)h(n)p
Ff(,)30 b(pr)l(ovides)i(an)d(e)l(quivalenc)l(e)i(b)l(etwe)l(en)e(the)h
(c)l(ate)l(gories.)100 3815 y Ft(In)40 b(addition)g(to)f(the)i(functor)
e Fr(S)5 b Ft(,)43 b(let)d(us)g(also)f(in)n(tro)r(duce,)k(for)c(the)h
(algebras)e Fv(P)2773 3828 y Fs(n;)r(~)-36 b(\013)2881
3815 y Ft(,)43 b(a)0 3915 y(functor)27 b Fr(T)34 b Ft(:)50
b Fr(R)q(ep)26 b Fv(P)673 3928 y Fs(n;)r(~)-36 b(\013)804
3915 y Fv(\000)-14 b(!)23 b Fr(R)q(ep)j Fv(P)1190 3938
y Fs(n;)1302 3916 y Fi(~)-33 b(\013)p 1261 3925 117 3
v 1261 3958 a(A)p Fh(\000)p Fd(1)1391 3915 y Ft(,)27
b(where)g Fr(A)c Ft(=)1854 3852 y Fn(P)1941 3873 y Fs(n)1941
3940 y(i)p Fq(=1)2067 3915 y Fr(\013)2120 3927 y Fs(i)2174
3915 y Ft(and)k Fr(A)d(>)e Ft(1.)36 b(Let)28 b Fr(\031)i
Ft(b)r(e)d(a)0 4036 y(represen)n(tation)g(of)i(the)h(algebra)d
Fv(P)1135 4049 y Fs(n;)r(~)-36 b(\013)1243 4036 y Ft(,)30
b Fr(\031)s Ft(\()p Fr(p)1420 4048 y Fs(i)1448 4036 y
Ft(\))c(=)f Fr(P)1649 4048 y Fs(i)1706 4036 y Ft(orthogonal)i(pro)5
b(jections)28 b(in)h(the)h(space)0 4142 y Fr(H)69 4154
y Fq(0)106 4142 y Ft(,)37 b(and)335 4079 y Fn(P)422 4100
y Fs(n)422 4167 y(i)p Fq(=1)548 4142 y Fr(\013)601 4154
y Fs(i)628 4142 y Fr(P)681 4154 y Fs(i)744 4142 y Ft(=)e
Fr(I)880 4154 y Fs(H)934 4162 y Fd(0)971 4142 y Ft(.)59
b(Let)1227 4121 y(\024)1209 4142 y Fr(P)1262 4154 y Fs(i)1325
4142 y Ft(=)34 b Fr(I)1460 4154 y Fs(H)1514 4162 y Fd(0)1575
4142 y Fv(\000)23 b Fr(P)1716 4154 y Fs(i)1744 4142 y
Ft(.)58 b(Then)2049 4079 y Fn(P)2137 4100 y Fs(n)2137
4167 y(i)p Fq(=1)2262 4142 y Fr(\013)2315 4154 y Fs(i)2343
4142 y Ft(\()p Fr(I)2411 4154 y Fs(H)2465 4162 y Fd(0)2526
4142 y Fv(\000)2632 4121 y Ft(\024)2614 4142 y Fr(P)2667
4154 y Fs(i)2695 4142 y Ft(\))35 b(=)f Fr(I)2897 4154
y Fs(H)2951 4162 y Fd(0)0 4248 y Ft(or)104 4186 y Fn(P)192
4206 y Fs(n)192 4273 y(i)p Fq(=1)360 4214 y Fs(\013)403
4222 y Fi(i)p 328 4229 135 4 v 328 4277 a Fs(A)p Fp(\000)p
Fq(1)491 4227 y Ft(\024)472 4248 y Fr(P)525 4260 y Fs(i)581
4248 y Ft(=)27 b Fr(I)709 4260 y Fs(H)763 4268 y Fd(0)801
4248 y Ft(.)45 b(Set)31 b Fr(T)12 b Ft(\()p Fr(\031)s
Ft(\))27 b(=)32 b(\024)-47 b Fr(\031)t Ft(,)31 b(where)j(\024)-46
b Fr(\031)s Ft(\()p Fr(p)1781 4260 y Fs(i)1809 4248 y
Ft(\))28 b(=)1979 4227 y(\024)1961 4248 y Fr(P)2014 4260
y Fs(i)2042 4248 y Ft(.)45 b Fr(T)12 b Ft(\()p Fr(\031)s
Ft(\))30 b(is)h(a)f(represen)n(tation)0 4350 y(of)37
b(the)g(algebra)e Fv(P)614 4373 y Fs(n;)726 4351 y Fi(~)-33
b(\013)p 685 4360 117 3 v 685 4393 a(A)p Fh(\000)p Fd(1)815
4350 y Ft(.)65 b(If)38 b Fr(C)1055 4362 y Fq(0)1129 4350
y Ft(is)f(a)g(morphism)f(from)h(the)g(represen)n(tation)e
Fr(\031)41 b Ft(in)n(to)36 b(the)0 4471 y(represen)n(tation)24
b(~)-47 b Fr(\031)t Ft(,)22 b(then)g(this)f(op)r(erator)e
Fr(C)1357 4483 y Fq(0)1416 4471 y Ft(\(the)j(represen)n(tation)d(space)
i(remains)f(the)h(same\))0 4570 y(de\014nes)32 b(a)g(morphism)f(from)h
(the)g(represen)n(tation)e Fr(T)12 b Ft(\()p Fr(\031)s
Ft(\))32 b(in)n(to)g(the)g(represen)n(tation)f Fr(T)12
b Ft(\()t(~)-46 b Fr(\031)r Ft(\))33 b(in)0 4670 y(suc)n(h)24
b(a)g(w)n(a)n(y)f(that)i Fr(T)12 b Ft(\()p Fr(C)744 4682
y Fq(0)781 4670 y Ft(\))23 b(=)g Fr(C)983 4682 y Fq(0)1020
4670 y Ft(.)36 b(It)25 b(is)f(clear)g(that)h Fr(T)35
b Ft(is)24 b(an)h(equiv)-5 b(alence)24 b(\(the)h(algebras)d
Fv(P)2881 4683 y Fs(n;)r(~)-36 b(\013)0 4770 y Ft(and)24
b Fv(P)216 4793 y Fs(n;)328 4771 y Fi(~)-33 b(\013)p
287 4780 V 287 4813 a(A)p Fh(\000)p Fd(1)441 4770 y Ft(are)23
b(isomorphic\).)35 b(Finally)-7 b(,)25 b(let)g(us)f(in)n(tro)r(duce)g
(Co)n(xeter)f(functors)h(\010+)g(and)g(\010)2933 4740
y Fp(\000)0 4906 y Ft(as)19 b(follo)n(ws:)32 b(\010)455
4876 y Fq(+)534 4906 y Ft(=)22 b Fr(S)5 b(T)34 b Ft(:)43
b Fr(R)q(ep)20 b Fv(P)1049 4919 y Fs(n;)r(~)-36 b(\013)1179
4906 y Fv(\000)-14 b(!)23 b(P)1394 4931 y Fs(n;)1450
4919 y(~)1455 4931 y Fq(1)p Fp(\000)1591 4909 y Fi(~)-33
b(\013)p 1550 4918 V 1550 4951 a(A)p Fh(\000)p Fd(1)1680
4906 y Ft(,)22 b(where)d Fr(A)24 b Ft(=)2130 4843 y Fn(P)2218
4864 y Fs(n)2218 4931 y(i)p Fq(=1)2343 4906 y Fr(\013)2396
4918 y Fs(i)2424 4906 y Ft(,)2462 4888 y Fr(~)2469 4906
y Ft(1)s Fv(\000)2639 4873 y Fs(~)-36 b(\013)p 2592 4887
135 4 v 2592 4934 a(A)p Fp(\000)p Fq(1)2759 4906 y Ft(=)23
b(\(1)s Fv(\000)40 4995 y Fs(\013)83 5003 y Fd(1)p 10
5010 V 10 5057 a Fs(A)p Fp(\000)p Fq(1)155 5029 y Fr(;)14
b Ft(1)8 b Fv(\000)352 4995 y Fs(\013)395 5003 y Fd(2)p
323 5010 V 323 5057 a Fs(A)p Fp(\000)p Fq(1)468 5029
y Fr(;)14 b(:)g(:)g(:)f(;)h Ft(1)8 b Fv(\000)809 4995
y Fs(\013)852 5003 y Fi(n)p 783 5010 V 783 5057 a Fs(A)p
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b(we)h(have)971 2141 y Fn(P)1059 2228 y Fs(i)p Fp(2)p
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y Fs(n)1207 2845 y Ft(for)43 b(an)n(y)h(represen)n(tation)e
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(to)g(b)l(e)0 4152 y Fr(\013)53 4164 y Fs(n)p Fq(+1)182
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4164 y Fs(n)p Fq(+)p Fs(m)614 4152 y Ff(.)41 b(Then)903
4130 y Fr(~)898 4152 y(\014)29 b Fv(2)c Fm(J)1100 4164
y Fs(n)p Fq(+)p Fs(m)1285 4152 y Ff(if)32 b(and)f(only)g(if)h(the)e(ve)
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4152 y Fr(;)g(:)g(:)g(:)f(;)h(\013)2831 4164 y Fs(n)2876
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4241 y Fr(~)1144 4262 y(\014)t Ff(.)38 b(F)-6 b(or)27
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4291 y(\014)2963 4262 y Ff(,)0 4373 y(we)i(have)h Fr(\031)s
Ft(\()p Fr(p)438 4385 y Fs(i)466 4373 y Ft(\))23 b(=)g(0)29
b Ff(for)i Fr(i)22 b Ft(=)h Fr(n)18 b Ft(+)g(1)p Fr(;)c(n)k
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4528 y(Pr)l(o)l(of.)44 b Ft(Using)30 b(Prop)r(osition)e(5)i(it)g(is)g
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4528 y(\014)j Fv(2)27 b Fm(J)2242 4540 y Fs(n)p Fq(+)p
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y(Let)23 b Fr(\031)k Ft(b)r(e)c(a)g(represen)n(tation)e(of)i(the)h
(algebra)d Fv(P)1502 4662 y Fs(n)p Fq(+)p Fs(m;)1677
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y Ft(\))g(=)e Fr(I)7 b Ft(,)24 b(it)0 4759 y(follo)n(ws)h(that)450
4697 y Fn(P)538 4718 y Fs(n)p Fq(+)p Fs(m)538 4784 y(i)p
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4759 y Fr(\013)866 4771 y Fs(i)894 4759 y Fr(\031)s Ft(\()p
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Fv(\000)15 b Fr(\013)1381 4771 y Fs(i)1404 4779 y Fd(0)1442
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y Fs(i)1876 4779 y Fd(0)1936 4759 y Fr(>)22 b Ft(1,)27
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4961 y(only)d(if)h Fr(\031)s Ft(\()p Fr(p)382 4973 y
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b(Th)n(us,)921 4899 y Fn(P)1008 4919 y Fs(n)1008 4986
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%%Page: 8 8
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Fv(2)c Fm(J)210 683 y Fs(n)p Fq(+)p Fs(m)365 671 y Ff(,)34
b(if)g Fr(B)f(>)c Ft(1)p Ff(,)k(the)g(ve)l(ctor)1193
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V 1055 822 a Fq(1)p Fp(\000)p Fs(B)1203 794 y Fr(;)14
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1397 775 V 1397 822 a Fq(1)p Fp(\000)p Fs(B)1545 702
y Fn(\021)1618 794 y Fv(2)23 b Fm(J)1742 806 y Fs(n)1787
794 y Ff(.)39 b(In)28 b(the)h(latter)g(c)l(ase,)h Fr(\031)s
Ft(\()p Fr(p)2643 806 y Fs(i)2671 794 y Ft(\))23 b(=)g
Fr(I)36 b Ff(for)0 914 y(a)30 b(r)l(epr)l(esentation)g
Fr(\031)j Ff(of)e(the)f(algebr)l(a)h Fv(P)1268 942 y
Fs(n)p Fq(+)p Fs(m;)1443 927 y(~)1439 942 y(\014)1513
914 y Ff(for)f Fr(i)23 b Ft(=)g Fr(n)18 b Ft(+)g(1)p
Fr(;)c(:)g(:)g(:)f(;)h(n)k Ft(+)g Fr(m)p Ff(.)0 1125
y(Pr)l(o)l(of.)44 b Ft(W)-7 b(e)26 b(ha)n(v)n(e)f Fr(T)12
b Ft(\()690 1103 y Fr(~)685 1125 y(\014)t Ft(\))23 b(=)879
1033 y Fn(\020)968 1091 y Fs(\013)1011 1099 y Fd(1)p
938 1106 135 4 v 938 1154 a Fs(A)p Fp(\000)p Fq(1)1083
1125 y Fr(;)1160 1091 y Fs(\013)1203 1099 y Fd(2)p 1130
1106 V 1130 1154 a Fs(A)p Fp(\000)p Fq(1)1275 1125 y
Fr(;)14 b(:)g(:)g(:)f(;)1469 1084 y Fs(\013)1512 1092
y Fi(n)p Fd(+)p Fi(m)p 1469 1106 178 4 v 1490 1154 a
Fs(A)p Fp(\000)p Fq(1)1657 1033 y Fn(\021)1732 1125 y
Ft(and)1934 1091 y Fs(\013)1977 1099 y Fi(i)p 1902 1106
135 4 v 1902 1154 a Fs(A)p Fp(\000)p Fq(1)2070 1125 y
Fr(>)22 b Ft(1)k(for)f Fr(i)e Ft(=)f Fr(n)15 b Ft(+)f(1)p
Fr(;)g(:)g(:)g(:)g(;)g(n)g Ft(+)0 1269 y Fr(m)p Ft(.)43
b(Then,)31 b(if)466 1247 y Fr(~)460 1269 y(\014)g Fv(2)c
Fm(J)666 1281 y Fs(n)p Fq(+)p Fs(m)851 1269 y Ft(and)j
Fr(\031)j Ft(is)d(a)f(represen)n(tation)f(of)i(the)g(algebra)e
Fv(P)2390 1297 y Fs(n)p Fq(+)p Fs(m;)2565 1282 y(~)2561
1297 y(\014)2605 1269 y Ft(,)j(it)f(follo)n(ws)0 1379
y(from)d(Prop)r(osition)f(8)i(that)g Fr(T)12 b Ft(\()p
Fr(\031)s Ft(\)\()p Fr(p)1142 1391 y Fs(i)1169 1379 y
Ft(\))24 b(=)f(0)k(for)g Fr(i)c Ft(=)g Fr(n)18 b Ft(+)g(1)p
Fr(;)c(:)g(:)g(:)f(;)h(n)19 b Ft(+)f Fr(m)27 b Ft(and,)h(hence,)g
Fr(\031)s Ft(\()p Fr(p)2841 1391 y Fs(i)2869 1379 y Ft(\))23
b(=)0 1479 y Fr(I)46 b Ft(for)39 b Fr(i)j Ft(=)g Fr(n)26
b Ft(+)g(1)p Fr(;)14 b(:)g(:)g(:)f(;)h(n)26 b Ft(+)g
Fr(m)p Ft(.)71 b(Then)1355 1416 y Fn(P)1442 1437 y Fs(n)1442
1504 y(i)p Fq(=1)1568 1479 y Fr(\013)1621 1491 y Fs(i)1649
1479 y Fr(\031)s Ft(\()p Fr(p)1773 1491 y Fs(i)1801 1479
y Ft(\))42 b(=)g(\(1)26 b Fv(\000)g Fr(B)t Ft(\))p Fr(I)7
b Ft(.)72 b(F)-7 b(or)39 b Fr(B)46 b(>)c Ft(1,)g(w)n(e)0
1578 y(get)c(a)g(con)n(tradiction,)i(since)f(the)g(op)r(erator)e(in)h
(the)h(left-hand)g(side)g(is)f(p)r(ositiv)n(e)g(and)g(the)0
1680 y(one)i(in)h(the)f(righ)n(t-hand)f(side)i(is)f(not.)75
b(If)41 b Fr(B)48 b(<)c Ft(1,)f(then)1971 1618 y Fn(P)2059
1639 y Fs(n)p Fq(+)p Fs(m)2059 1705 y(i)p Fq(=)p Fs(n)p
Fq(+1)2320 1646 y Fs(\013)2363 1654 y Fi(i)p 2286 1661
138 4 v 2286 1709 a Fq(1)p Fp(\000)p Fs(B)2434 1680 y
Fr(\031)s Ft(\()p Fr(p)2558 1692 y Fs(i)2586 1680 y Ft(\))i(=)f
Fr(I)j Ft(and)0 1728 y Fn(\020)91 1786 y Fs(\013)134
1794 y Fd(1)p 60 1801 V 60 1848 a Fq(1)p Fp(\000)p Fs(B)207
1820 y Fr(;)286 1786 y Fs(\013)329 1794 y Fd(2)p 254
1801 V 254 1848 a Fq(1)p Fp(\000)p Fs(B)402 1820 y Fr(;)14
b(:)g(:)g(:)g(;)624 1786 y Fs(\013)667 1794 y Fi(n)p
597 1801 V 597 1848 a Fq(1)p Fp(\000)p Fs(B)745 1728
y Fn(\021)831 1820 y Fv(2)37 b Fm(J)969 1832 y Fs(n)1014
1820 y Ft(.)61 b(If)37 b Fr(B)j Ft(=)d(1,)g(w)n(e)e(ha)n(v)n(e)1832
1798 y Fr(~)1827 1820 y(\014)41 b Fv(2)c Fm(J)2053 1832
y Fs(n)p Fq(+)p Fs(m)2244 1820 y Ft(using)e(Prop)r(osition)f(6.)0
1977 y(If)88 1885 y Fn(\020)179 1943 y Fs(\013)222 1951
y Fd(1)p 147 1958 V 147 2006 a Fq(1)p Fp(\000)p Fs(B)295
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2828 y Fs(i)1750 2816 y Fv(\024)g Ft(2)d(and)h(the)h(v)n(ector)f
Fr(~)-44 b(\013)38 b Ft(satisfy)e(the)0 2919 y Fr(R)q
Ft(-condition.)g(When)28 b(do)r(es)f(the)h(v)n(ector)h
Fr(~)-44 b(\013)28 b Ft(b)r(elong)f(to)g Fm(J)1803 2931
y Fs(n)1849 2919 y Ft(?)100 3018 y(In)h(this)h(pap)r(er)g(w)n(e)f(also)
f(w)n(ere)h(not)h(solving)e(the)i(problem)f(of)h(classifying,)f(up)h
(to)f(unitary)0 3118 y(equiv)-5 b(alence,)31 b(all)g(irreducible)f
(represen)n(tations)e(of)j(the)g(algebras)e Fv(P)2205
3131 y Fs(n;)r(~)-36 b(\013)2344 3118 y Ft(for)32 b Fr(~)-44
b(\013)29 b Fv(2)f Fm(J)2685 3130 y Fs(n)2730 3118 y
Ft(.)47 b(Ho)n(w-)0 3218 y(ev)n(er,)34 b(there)g(is)f(no)h(doubt)g
(that)g(the)g(constructiv)n(e)f(functors)g Fr(S)5 b Ft(,)35
b Fr(T)12 b Ft(,)35 b(\010)2310 3188 y Fq(+)2365 3218
y Ft(,)g(and)f(\010)2651 3188 y Fp(\000)2741 3218 y Ft(will)f(b)r(e)0
3317 y(useful)28 b(for)f(suc)n(h)g(a)h(classi\014cation.)1252
3492 y Fu(References)35 3625 y Fl(1.)35 b(Rabano)n(vic)n(h)26
b(V.)e(I.,)f(Samo)-8 b(\024)-27 b(\020lenk)n(o)25 b(Y)-6
b(u.)24 b(S.,)g Fk(When)i(sum)h(of)f(idemp)l(otents)h(or)f(ortho)l
(gonal)j(pr)l(oje)l(ctions)e(is)125 3708 y(a)f(multiple)h(of)e(unity)p
Fl(,)e(F)-6 b(unktsion.)24 b(Anal.)g(i)f(Prilozhen.)g
Fc(34)g Fl(\(2000\),)i(no.)f(4,)f(91{93.)32 b(\(Russian\))35
3791 y(2.)j(Rabano)n(vic)n(h)26 b(V.)c(I.,)h(Samo)-8
b(\024)-27 b(\020lenk)n(o)25 b(Y)-6 b(u.)23 b(S.,)g Fk(Sc)l(alar)k(op)l
(er)l(ators)g(r)l(epr)l(esentable)f(as)g(a)g(sum)g(of)g(pr)l(oje)l
(ctions)p Fl(,)125 3874 y(Ukrain.)d(Mat.)h(Zh.)f Fc(53)g
Fl(\(2001\),)i(no.)f(7,)f(939{952.)33 b(\(Russian\))35
3957 y(3.)i(Krugly)n(ak)21 b(S.)e(A.,)h Fk(Coxeter)i(functors)h(for)f
(a)h(c)l(ertain)f(class)h(of)g Fj(\003)p Fk(-quivers)p
Fl(,)c(Ukrain.)h(Mat.)g(Zh.)g Fc(54)g Fl(\(2002\),)125
4040 y(no.)k(6,)f(789-797.)33 b(\(Russian\))35 4123 y(4.)i(Bernsh)n
(tein)20 b(I.)g(N.,)f(Gelfand)h(I.)f(M.,)g(P)n(onamarev)h(V.)f(A.,)g
Fk(Coxeter)i(functors)h(and)g(a)g(the)l(or)l(em)h(of)f(Gabriel)p
Fl(,)125 4206 y(Usp)r(ekhi)i(Mat.)g(Nauk)g Fc(28)f Fl(\(1973\),)i(no.)f
(2,)f(19{33.)32 b(\(Russian\))35 4289 y(5.)j(Krugly)n(ak)19
b(S.)f(A.,)g(Rabano)n(vic)n(h)i(V.)d(I.,)i(Samo)-8 b(\024)-27
b(\020lenk)n(o)19 b(Y)-6 b(u.)18 b(S,)h Fk(On)h(sums)h(of)g(pr)l(oje)l
(ctions)p Fl(,)f(F)-6 b(unktsion.)19 b(Anal.)125 4372
y(i)24 b(Prilozhen.)f Fc(36)g Fl(\(2002\),)i(no.)e(3,)h(20{35.)32
b(\(Russian\))35 4455 y(6.)j(Galins'ky)27 b(D.)f(V.,)g(Krugly)n(ak)g
(S.)g(A.,)g Fk(R)l(epr)l(esentations)j(of)g Fj(\003)p
Fk(-algebr)l(as)g(gener)l(ate)l(d)f(by)g(line)l(arly)h(r)l(elate)l(d)
125 4538 y(ortho)l(gonal)f(pr)l(oje)l(ctions)p Fl(,)d(Visn)n(yk)e
(Kyivs'k)n(ogo)h(Univ)n(ersytetu)h(\(1999\),)g(no.)f(2,)f(24{31.)33
b(\(Ukrainian\))35 4621 y(7.)i(Roiter)22 b(A.)g(V.,)f
Fk(Boxes)j(with)h(involution)p Fl(,)d(Represen)n(tations)i(and)e
(quadratic)h(forms,)e(Institute)j(of)e(Math-)125 4704
y(ematics,)i(Academ)n(y)h(of)e(Sciences)i(of)f(USSR,)f(Kiev,)g(1979,)h
(pp.)g(124{126.)33 b(\(Russian\))100 4850 y Fb(Institute)h(of)g(Ma)l
(thema)l(tics,)i(Ukrainian)f(Na)l(tional)f(A)n(cademy)h(of)f(Sciences,)
i(3)e(Teresh-)0 4933 y(chenkivs'ka,)25 b(Kyiv,)i(01601,)c(Ukraine)100
5016 y Fk(E-mail)j(addr)l(ess)5 b Fl(:)37 b Fa(yurii)p
800 5016 22 4 v 27 w(sam@imath.kiev.ua)p eop end
%%Trailer

userdict /end-hook known{end-hook}if
%%EOF