Highest-weight vectors for the adjoint action of GL_n on polynomials, II

TL;DR

This paper extends the explicit description of highest-weight vectors for the invariants of polynomial functions on gl_n under conjugation, providing new bases for a broader class of weights using Cayley-Hamilton identities.

Contribution

It introduces explicit bases for modules of highest weight vectors for a larger class of weights, generalizing previous results with new Cayley-Hamilton type identities.

Findings

Explicit bases for modules of highest weight vectors are constructed.

The results apply to a broader class of weights than previously known.

Cayley-Hamilton type identities are proved to express semi-invariants.

Abstract

Let G=GL_n be the general linear group over an algebraically closed field k and let g=gl_n be its Lie algebra. Let U be the subgroup of G which consists of the upper unitriangular matrices. Let k[g] be the algebra of polynomial functions on g and let k[g]^G be the algebra of invariants under the conjugation action of G. For all weights chi in Z^n with chi_2<=0 or chi_{n-1}>=0 we give explicit bases for the k[g]^G-module k[g]^U_chi of highest weight vectors of weight chi. This extends earlier results to a much bigger class of weights. To express our semi-invariants in terms of matrix powers we prove certain Cayley-Hamilton type identities.