Bases for spaces of highest weight vectors in arbitrary characteristic

TL;DR

This paper provides explicit bases for highest weight vectors in coordinate rings of matrix spaces under group actions over algebraically closed fields of any characteristic, using filtrations and module decompositions.

Contribution

It introduces explicit bases and filtrations for highest weight vectors in matrix coordinate rings for arbitrary characteristic, extending prior characteristic-zero results.

Findings

Explicit bases for highest weight vectors in matrix coordinate rings.

A good GL_r x GL_s-filtration on the coordinate ring.

Explicit spanning sets for modules of highest weight vectors.

Abstract

Let k be an algebraically closed field of arbitrary characteristic. First we give explicit bases for the highest weight vectors for the action of GL_r x GL_s on the coordinate ring k[Mat_{rs}^m] of m-tuples of r x s-matrices. It turns out that this is done most conveniently by giving an explicit good GL_r x GL_s-filtration on k[Mat_{rs}^m]. Then we deduce from this result explicit spanning sets of the k[Mat_n]^{GL_n}-modules of highest weight vectors in the coordinate ring k[Mat_n] under the conjugation action of GL_n.