Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 091, 6 pages      arXiv:1708.07782      https://doi.org/10.3842/SIGMA.2017.091
Contribution to the Special Issue on the Representation Theory of the Symmetric Groups and Related Topics

James' Submodule Theorem and the Steinberg Module

Meinolf Geck
IAZ - Lehrstuhl für Algebra, Universität Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany

Received August 29, 2017, in final form November 28, 2017; Published online December 05, 2017

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.

Key words: groups with a $BN$-pair; Steinberg representation; modular representations.

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