Vertices of Low-Dimensional Simple Modules for Symmetric Groups

Abstract

We determine the vertices of all simple modules for the symmetric groups over an algebraically closed field of prime characteristic whose dimensions are at most 1000 and give combinatorial descriptions of both the modules and their vertices. Moreover, we formulate some general conjectures on vertices of several series of simple modules for symmetric groups, and provide some evidence for these to hold true.

Key Words:

• Simple module
• Symmetric group
• Vertex

2000 Mathematics Subject Classification:

• 20C20
• 20C30

ACKNOWLEDGMENTS

The results of this article are taken from the author's Ph.D. thesis (cf. Danz, 2007a) which had been supported by the “Deutsche Forschungsgemeinschaft” (DFG). The author is grateful to her supervisor Prof. Burkhard Külshammer and to Dr. Jürgen Müller for many valuable discussions and their support while she was working on her thesis. Furthermore, the author wishes to thank the referee for helpful comments on an earlier version of this article.

Notes

¹This has recently been provide by Külshammer and the author in Danz, S., Külshammer, B. The vertices and sources of the basic spin module for the symmetric group in characteristic 2. J. Pure and Applied Algebra. To appear.

Communicated by A. Turull.