weylModulesMap -- build a map between Weyl modules from its action on tableaux

Usage:

weylModulesMap(N, M, F)
Inputs:
- N, a module, a Weyl module from weylModule
- M, a module, a Weyl module from weylModule
- F, a function, a function such that for every Weyl-semistandard filling T in the basis of M, F(T) produces a list of pairs (c, T') with c in ring N and T' a (not-necessarily-standard) filling of the shape of N
Outputs:
- a matrix, the $R$-linear map M -> N whose value on each basis tableau is straightened into N

Description

The Weyl analogue of schurModulesMap. The shape of N is the target shape of the tableaux produced by F; the straightening into the Weyl-standard basis is performed internally using weylStraighten.

Example: a multiplication-style map $W_{(2)}(E) \to W_{(2,1)}(E) \otimes E^*$. The analogue of the Koszul differential, with divided-power multiplication:

i1 : n = 3;

i2 : R = QQ[x_1..x_n];

i3 : M = weylModule({2}, R^n);

i4 : N = weylModule({2,1}, R^n);

i5 : F = T -> apply(numgens R, k -> (R_k, augmentWeylFilling(T, 1, k)));

i6 : weylModulesMap(N, M, F)

o6 = | x_2 -x_1 0    0    0    0    |
     | x_3 0    0    -x_1 0    0    |
     | 0   x_2  -x_1 0    0    0    |
     | 0   x_3  0    0    -x_1 0    |
     | 0   0    x_3  0    -x_2 0    |
     | 0   0    0    x_2  -x_1 0    |
     | 0   0    0    x_3  0    -x_1 |
     | 0   0    0    0    x_3  -x_2 |

             8      6
o6 : Matrix R  <-- R

Ways to use weylModulesMap:

weylModulesMap(Module,Module,Function)

For the programmer

The object weylModulesMap is a method function.

The source of this document is in SchurFunctors.m2:2372:0.

weylModulesMap -- build a map between Weyl modules from its action on tableaux

Description

See also

Ways to use weylModulesMap:

For the programmer