
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, D70569 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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