add skew-symm matrix Schuberts

benchmarkIdeals.m2

@@ -32,9 +32,6 @@ time radical(I, Strategy=>CompleteIntersection);
 -- approx. avg. time for Strategy=>CompleteIntersection: at most 0.001s
 
 
-
-
-
 restart
 
 
@@ -69,7 +66,82 @@ time radical I
 
 
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--- kernel of a map from a polynomial ring k[x_1..x_n] to a ring of the form
+-- Skew-symmetric matrix Schubert varieties are exactly matrix Schubert varieties
+-- with the underlying generic matrix replaced by a generic skew-symmetric matrix.
+-- The ideal constructed in this way is in general not radical.
+-- Reference: Eric Marberg and Brendan Pawlowski. Groebner geometry for skew-symmetric
+--	       matrix Schubert varieties.
+------------------------------------------------------------------------------------------
+
+-- two remarks:
+-- 1) skew-symmetric matrix Schuberts only make sense for permutations that are
+--     fixed-point-free involutions, defined in the method isFPFInv
+-- 2) there are no fixed-point-free involutions in S_n for odd n, so we restrict to even n
+
+restart
+
+needsPackage "MatrixSchubert"
+
+isFPFInv = w -> (
+    -- returns true iff the permutation w (a list) is a Fixed-Point-Free Involution
+    -- def: w is fixed-point-free iff w(i) never equals i
+    -- def: w is an involution iff w^2 = id
+    -- so w is a fixed-point-free involution if its permutation matrix is symmetric with 0's on the diagonal
+    n := #w;
+    wMatrix := map(ZZ^n, n, (i,j) -> if w#j == i+1 then 1 else 0);
+
+    (wMatrix == transpose wMatrix) and not any(w, i -> i == w#(i-1))
+    )
+
+
+skewSymmEssentialSet = w -> (
+    -- the usual essential set, but only the (i, j) for which i < j
+    D := select(rotheDiagram w, L -> (L_0 < L_1));  -- the skew-symmetric diagram
+    select(D, L -> ( not( isMember((L_0+1, L_1), D) or  isMember((L_0, L_1+1), D) )))
+    )
+
+
+skewSymmDetIdeal = (R, w) -> (
+    -- returns the skew-symmetric matrix Schubert determinantal ideal
+    -- the radical of this ideal defines the skew-symmetric matrix Schubert variety
+
+    n := #w;
+    M := genericSkewMatrix(R, n);
+
+    -- get the essential set and re-index 
+    essSet := apply(skewSymmEssentialSet w, L -> (L_0 - 1, L_1 - 1));
+
+    -- find the determinantal ideal
+    ranks := rankTable w;
+    ideal apply(essSet, L -> (
+	    row := L_0;
+	    col := L_1;
+	    r := ranks_(row, col);
+
+	    minors(r+1, M^(toList(0..row))_(toList(0..col)))
+	    )
+        )
+    )
+
+
+R = QQ[x_1..x_100];
+n = 8;  -- n must be even
+Sn = permutations toList (1..n);
+fpfInvs = select(Sn, isFPFInv);
+
+for w in fpfInvs do (
+    I := skewSymmDetIdeal(R, w);
+    time radical(I, Strategy=>Decompose);
+    )
+
+
+-- avg. approx. time for Strategy=>Decompose: most are a fraction of a second (between 0.1s and 0.5s), some are quicker, a couple are 20-30s
+-- assumptions not met for CompleteIntersection method
+
+
+
+------------------------------------------------------------------------------------------
+-- KERNEL of a map from a polynomial ring k[x_1..x_n] to a ring of the form
 -- k[t^S, x^[p]], where S is a numerical semigroup and x^[p] is the ideal of pth-powers
 -- of some variables
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