decomposeRep -- decompose a polynomial GL-representation into irreducible subspaces

Usage:

decomposeRep F
Inputs:
- F, a matrix, a square matrix representing a polynomial $GL_d$-action, as in characterRep
Outputs:
- a hash table, keyed by the partitions $\lambda$ appearing in the decomposition of F (as returned by splitCharacter), with each value a matrix whose columns span the $\lambda$-isotypic subspace

Description

Given a polynomial $GL_d$-representation $W$ whose matrix is F, decomposeRep computes the character via characterRep, decomposes it via splitCharacter, and for each irreducible $S_\lambda$ appearing produces a basis of the $S_\lambda$-isotypic subspace as the syzygy space of two operators:

the transvections (upper-triangular unipotent generators) must act trivially on a highest-weight vector;
the diagonal torus must act by the maximal-weight character of the $\lambda$-isotypic piece.

The partitions appearing as keys, and their multiplicities (as the number of columns of the corresponding basis matrix), agree with the output of splitCharacter characterRep F.

i1 : R = QQ[w_1..w_9];

i2 : G = genericMatrix(R, 3, 3);

             3      3
o2 : Matrix R  <-- R

i3 : H = decomposeRep schur({2}, schur({2}, G));

i4 : keys H

o4 = {{2, 2}, {4}}

o4 : List

Ways to use decomposeRep:

decomposeRep(Matrix)

For the programmer

The object decomposeRep is a method function.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SchurFunctors.m2:2776:0.

decomposeRep -- decompose a polynomial GL-representation into irreducible subspaces

Description

See also

Ways to use decomposeRep:

For the programmer