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The group of endotrivial modules for the symmetric and alternating groups

Part of: Representation theory of groups

Published online by Cambridge University Press:  12 January 2010

Jon F. Carlson ,
David J. Hemmer  and
Nadia Mazza
Jon F. Carlson
Affiliation:
Department of Mathematics, University of Georgia, Athens, GA 30602, USA
David J. Hemmer
Affiliation:
Department of Mathematics, University at Buffalo SUNY, 244 Mathematics Building, Buffalo, NY 14260, USA, Email: (dhemmer@math.buffalo.edu)
Nadia Mazza
Affiliation:
Department of Mathematical Sciences, University of Aberdeen, Aberdeen AB24 3UE, UK
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Abstract

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We complete a classification of the groups of endotrivial modules for the modular group algebras of symmetric groups and alternating groups. We show that, for n ≥ p2, the torsion subgroup of the group of endotrivial modules for the symmetric groups is generated by the sign representation. The torsion subgroup is trivial for the alternating groups. The torsion-free part of the group is free abelian of rank 1 if n ≥ p2 + p and has rank 2 if p2 ≤ n < p2 + p. This completes the work begun earlier by Carlson, Mazza and Nakano.

Keywords

endotrivial moduleendopermutation modulesymmetric groupsalternating groups

MSC classification

Secondary: 20C20: Modular representations and characters
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 53 , Issue 1 , February 2010 , pp. 83 - 95
Copyright
Copyright © Edinburgh Mathematical Society 2010