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Rational representations and permutation representations of finite groups

Mathematische Annalen volume 364, pages 539–558 (2016)Cite this article

Abstract

We investigate the question which \(\mathbb {Q}\)-valued characters and characters of \(\mathbb {Q}\)-representations of finite groups are \(\mathbb {Z}\)-linear combinations of permutation characters. This question is known to reduce to that for quasi-elementary groups, and we give a solution in that case. As one of the applications, we exhibit a family of simple groups with rational representations whose smallest multiple that is a permutation representation can be arbitrarily large.

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Acknowledgments

The first author is supported by a Research Fellowship from the Royal Commission for the Exhibition of 1851, and the second author is supported by a Royal Society University Research Fellowship. We would like to thank Alexandre Turull for his help with Corollary 6.6. We are grateful to an anonymous referee for a careful reading of the manuscript and many helpful comments.

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Authors and Affiliations

  1. Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK

    Alex Bartel

  2. Department of Mathematics, University Walk, Bristol, BS8 1TW, UK

    Tim Dokchitser

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  1. Alex Bartel
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  2. Tim Dokchitser
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Correspondence to Alex Bartel.

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Bartel, A., Dokchitser, T. Rational representations and permutation representations of finite groups. Math. Ann. 364, 539–558 (2016). https://doi.org/10.1007/s00208-015-1223-y

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