James' Submodule Theorem and the Steinberg Module

TL;DR

This paper extends James' submodule theorem to finite groups with split BN-pairs, identifying a key composition factor of the Steinberg module and highlighting an open problem regarding its dimension.

Contribution

It generalizes James' submodule theorem to a broader class of finite groups with split BN-pairs and describes a distinguished composition factor of the Steinberg module.

Findings

Identifies a distinguished composition factor of the Steinberg module.

Extends the submodule theorem to groups with split BN-pairs.

Highlights an open problem on the dimension of this composition factor.

Abstract

James' submodule theorem is a fundamental result in the representation theory of the symmetric groups and the finite general linear groups. In this note we consider a version of that theorem for a general finite group with a split $BN$-pair. This gives rise to a distinguished composition factor of the Steinberg module, first described by Hiss via a somewhat different method. It is a major open problem to determine the dimension of this composition factor.