### 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.

pdf (325 kb)   tex (12 kb)

References

1. Carter R.W., Finite groups of Lie type. Conjugacy classes and complex characters, Wiley Classics Library, John Wiley & Sons, Ltd., Chichester, 1993.
2. Digne F., Michel J., Representations of finite groups of Lie type, London Mathematical Society Student Texts, Vol. 21, Cambridge University Press, Cambridge, 1991.
3. Geck M., On the modular composition factors of the Steinberg representation, J. Algebra 475 (2017), 370-391.
4. Gow R., The Steinberg lattice of a finite Chevalley group and its modular reduction, J. London Math. Soc. 67 (2003), 593-608.
5. Hiss G., Harish-Chandra series of Brauer characters in a finite group with a split $BN$-pair, J. London Math. Soc. 48 (1993), 219-228.
6. James G.D., Representations of general linear groups, London Mathematical Society Lecture Note Series, Vol. 94, Cambridge University Press, Cambridge, 1984.
7. Landrock P., Michler G.O., Principal $2$-blocks of the simple groups of Ree type, Trans. Amer. Math. Soc. 260 (1980), 83-111.
8. Steinberg R., Prime power representations of finite linear groups. II, Canad. J. Math. 9 (1957), 347-351.
9. Szechtman F., Steinberg lattice of the general linear group and its modular reduction, J. Group Theory 14 (2011), 603-635, arXiv:0812.2232.
10. Tinberg N.B., The Steinberg component of a finite group with a split $(B,N)$-pair, J. Algebra 104 (1986), 126-134.