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Endotrivial modules for the general linear group in a nondefining characteristic

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Abstract

Suppose that \(G\) is a finite group such that \(\mathrm{SL }(n,q)\subseteq G \subseteq \mathrm{GL }(n,q)\), and that \(Z\) is a central subgroup of \(G\). Let \(T(G/Z)\) be the abelian group of equivalence classes of endotrivial \(k(G/Z)\)-modules, where \(k\) is an algebraically closed field of characteristic \(p\) not dividing \(q\). We show that the torsion free rank of \(T(G/Z)\) is at most one, and we determine \(T(G/Z)\) in the case that the Sylow \(p\)-subgroup of \(G\) is abelian and nontrivial. The proofs for the torsion subgroup of \(T(G/Z)\) use the theory of Young modules for \(\mathrm{GL }(n,q)\) and a new method due to Balmer for computing the kernel of restrictions in the group of endotrivial modules.

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Acknowledgments

The authors would like to thank Paul Balmer for many helpful conversations and for urging us to look closely at his papers on the kernel of restriction. The second author acknowledges the Department of Mathematics at the University of Georgia for its support and hospitality during the initial stages of this project. The authors of this paper drew a lot of inspiration from experimental calculations using the computer algebra system Magma [5], though none of it is actually used in the proofs of this paper. We are grateful to Caroline Lassueur for pointing out an error in an earlier version of the paper. Finally, we wish to thank the referee for many helpful suggestions.

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Authors and Affiliations

  1. Department of Mathematics, University of Georgia, Athens, GA, 30602, USA

    Jon F. Carlson & Daniel K. Nakano

  2. Department of Mathematics, University of Lancaster, Lancaster, UK

    Nadia Mazza

Authors
  1. Jon F. Carlson
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  2. Nadia Mazza
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  3. Daniel K. Nakano
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Corresponding author

Correspondence to Nadia Mazza.

Additional information

Research of the first author was supported in part by NSF Grant DMS-1001102.

Research of the third author was supported in part by NSF Grant DMS-1002135.

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Carlson, J.F., Mazza, N. & Nakano, D.K. Endotrivial modules for the general linear group in a nondefining characteristic. Math. Z. 278, 901–925 (2014). https://doi.org/10.1007/s00209-014-1338-y

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  • DOI: https://doi.org/10.1007/s00209-014-1338-y

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