Twisted Dickson-Mui invariants and the Steinberg module multiplicity

TL;DR

This paper computes invariants and Steinberg module multiplicities for finite general linear groups acting on tensor products of symmetric and exterior algebras, extending understanding of their representation structure.

Contribution

It provides explicit formulas for invariants with arbitrary determinant twists and the graded multiplicity of the Steinberg module in these tensor products.

Findings

Determined invariants with arbitrary determinant twists for GL_n(q)

Calculated the graded multiplicity of the Steinberg module

Extended known results to include tensor products with twists

Abstract

We determine the invariants, with arbitrary determinant twists, of the parabolic subgroups of the finite general linear group GL_n(q) acting on the tensor product of the symmetric algebra and the exterior algebra of the natural GL_n(q)-module V. In addition, we obtain the graded multiplicity of the Steinberg module of GL_n(q) in the tensor product of the symmetric algebra and the exterior algebra of V, twisted by an arbitrary determinant power.