The GL_n(q)-module structure of the symmetric algebra around the Steinberg module

TL;DR

This paper analyzes the structure of the symmetric algebra of the natural GL_n(q)-module, focusing on the composition multiplicities around the Steinberg module, using algebraic group connections.

Contribution

It determines the graded composition multiplicities in the symmetric algebra for a broad class of modules near the Steinberg module, advancing understanding of module structures.

Findings

Computed Steinberg module multiplicities in tensor products.

Identified composition multiplicities in the symmetric algebra.

Connected algebraic group theory to module structure analysis.

Abstract

We determine the graded composition multiplicity in the symmetric algebra S(V) of the natural GL_n(q)-module V, or equivalently in the coinvariant algebra of V, for a large class of irreducible modules around the Steinberg module. This was built on a computation, via connections to algebraic groups, of the Steinberg module multiplicity in a tensor product of S(V) with other tensor spaces of fundamental weight modules.