schurModulesMap -- build a map between Schur modules from its action on tableaux

Usage:

schurModulesMap(N, M, F)
Inputs:
- N, a module, a Schur module from schurModule
- M, a module, a Schur module from schurModule
- F, a function, a function describing the map on tableaux: for every SSYT T in the basis of M, F(T) should produce a list of pairs (c, T') where c is a coefficient in ring N and T' is a (not-necessarily-semistandard) filling of the shape of N
Outputs:
- a matrix, the $R$-linear map M -> N given, on each basis tableau T of M, by the $R$-linear combination $\sum (c, T') \in F(T)$, implicitly straightened into $N$

Description

This constructor is the main tool for producing Schur-module homomorphisms from combinatorial data. It amounts to specifying where each tableau in a basis of the source goes, in a possibly-non-standard form; the straightening algorithm is invoked internally on each output to assemble the resulting vector in N.

Example: a piece of a Koszul-type differential. For $\lambda = (1^j)$ and $\lambda' = (1^{j+1})$ the natural map $\wedge^j(E) \to \wedge^{j+1}(E) \otimes E^*$ sends a basis tableau to $\sum_k x_k \otimes (T \cup \{k\})$. Lifting this via schurModulesMap gives (a piece of) the Koszul differential:

i1 : n = 4;

i2 : j = 2;

i3 : R = QQ[x_1..x_n];

i4 : M = schurModule(toList(j : 1), R^n);

i5 : N = schurModule(toList(j+1 : 1), R^n);

i6 : F = T -> apply(numgens R, k -> (R_k, augmentFilling(T, 0, k)));

i7 : schurModulesMap(N, M, F)

o7 = | x_3 -x_2 x_1 0    0    0   |
     | x_4 0    0   -x_2 x_1  0   |
     | 0   x_4  0   -x_3 0    x_1 |
     | 0   0    x_4 0    -x_3 x_2 |

             4      6
o7 : Matrix R  <-- R

Ways to use schurModulesMap:

schurModulesMap(Module,Module,Function)

For the programmer

The object schurModulesMap is a method function.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SchurFunctors.m2:2142:0.

schurModulesMap -- build a map between Schur modules from its action on tableaux

Description

See also

Ways to use schurModulesMap:

For the programmer