Decomposition of tensor products involving a Steinberg module

TL;DR

This paper provides formulas for decomposing tensor products involving Steinberg modules across algebraic groups, Frobenius kernels, and finite groups, introduces a character-based bilinear form, and explores tilting and simple module reciprocity related to Donkin's conjecture.

Contribution

It introduces a new character-based bilinear form, derives decomposition formulas, and links tilting module reciprocity to Donkin's tilting conjecture.

Findings

Formulas for tensor product decompositions involving Steinberg modules.

A bilinear form on characters used to compute homomorphism space dimensions.

A new proof of tilting and simple module reciprocity, related to Donkin's conjecture.

Abstract

We study the decomposition of tensor products between a Steinberg module and a costandard module, both as a module for the algebraic group $G$ and when restricted to either a Frobenius kernel $G_r$ or a finite Chevalley group $G(\mathbb{F}_q)$. In all three cases, we give formulas reducing this to standard character data for $G$. Along the way, we define a bilinear form on the characters of finite dimensional $G$-modules and use this to give formulas for the dimension of homomorphism spaces between certain $G$-modules when restricted to either $G_r$ or $G(\mathbb{F}_q)$. Further, this form allows us to give a new proof of the reciprocity between tilting modules and simple modules for $G$ which has slightly weaker assumptions than earlier such proofs. Finally, we prove that in a suitable formulation, this reciprocity is equivalent to Donkin's tilting conjecture.