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-- In-Reply-To: <200610201925.k9KJPC98001554@u123.math.uiuc.edu> -- Cc: bernd@Math.Berkeley.EDU, mike@math.cornell.edu, degraaf@science.unitn.it -- Subject: Re: equations for Schubert varieties -- Date: Sat, 28 Oct 2006 00:09:17 -0700 -- To: dan@math.uiuc.edu -- From: Josephine Yu <josephine.yu@gmail.com> -- -- Dear Dan, -- -- I think the code is fine as it is now. I'm pretty sure we want to -- keep only the variables that are <= the given one. Well of course, -- we didn't define what the Schubert variety "indexed by" a set really -- is. (It seems more common for them to be indexed by Young diagrams, -- or partitions. Maybe someone, or I, can implement that at some -- point. It won't be very hard.) -- -- Here in this example, the Schubert variety indexed by sigma = {1,2,4} -- in Gr(2,4) is the set of 3-planes in K^5 that intersects V_(sigma_i) -- at dimension at least i, where V_j is spanned by e_1, ... , e_j. So -- we surely want to keep the variable p_{1,2,4} and not throw it in the -- ideal. And p_tau = 0 if tau is not <= sigma, so p_tau should be in -- the ideal. -- -- Josephine -- -- On Oct 20, 2006, at 12:25 PM, Dan Grayson wrote: -- -- > -- > Bernd, -- > -- > I think it's throwing in the wrong variables. I've looked at the code, and it -- > throws in the variables whose tuples are not <= the given one in the product -- > ordering. This keeps the variables whose tuples that are <= the given one. -- > Since (2,3,4), (1,3,4), and (0,3,4) are greater than (1,2,4) (in the middle -- > spot) it throws the corresponding variables in. -- > -- > i1 : I = Schubert(2,4,{1,2,4}) -- > -- > o1 = ideal (p , p , p , p p - p p , p p - p p , p p - p p ) -- > 2,3,4 1,3,4 0,3,4 1,2,3 0,2,4 0,2,3 1,2,4 1,2,3 0,1,4 0,1,3 1,2,4 0,2,3 0,1,4 0,1,3 0,2,4 -- > -- > o1 : Ideal of ZZ [p , p , p , p , p , p , p , p , p , p ] -- > 0,1,2 0,1,3 0,2,3 1,2,3 0,1,4 0,2,4 1,2,4 0,3,4 1,3,4 2,3,4 -- > -- > It seems from your remarks that it should really be throwing in all variables -- > whose tuples are greater than or equal to {1,2,4} in the product ordering. -- > That retains the variables that are not >= {1,2,4}. -- this was the proposed change -- --- schubert.m2 (revision 4731) -- +++ schubert.m2 (working copy) -- @@ -49,7 +49,7 @@ -- L := toSequence \ subsets(n+1,k+1); -- R := o.CoefficientRing (monoid [apply(L, i -> new IndexedVariable from {baseName o.Variable,unsequence i})]); -- vr := new HashTable from apply(#L, i -> L#i => R_i); -- - higher := apply( select( L, s -> any(s, sigma, (a,b) -> a>b)), s -> vr#s ); -- + higher := apply( select( L, s -> all(s, sigma, (a,b) -> a>=b)), s -> vr#s ); -- G := flatten for i from 0 to #L-1 list for j from i+1 to #L-1 list ( -- r := L#i; -- s := L#j; I = Schubert(2,4,sigma = {1,2,4}) use ((ring I)/I) assert ( p_(1,2,4) != 0 ) assert all( toSequence \ subsets( {0,1,2,3,4}, 3 ), tau -> if all(tau,sigma,(i,j)->i<=j) then p_tau != 0 else p_tau == 0)