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Unisingular representations in arithmetic and Lie theory

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Abstract

Let G be a subgroup of \(\hbox {GL}(V)\), where V is a finite dimensional vector space over a finite field of characteristic \(p >0\). If \(\det (g-1) = 0\) for all \(g \in G\) then we call G a fixed-point subgroup of \(\hbox {GL}(V)\). Motivated in parallel by questions in arithmetic and linear group theory, we classify all irreducible fixed-point subgroups of \({\text {Sp}}_8(2)\) and give new infinite series of irreducible fixed-point subgroups of symplectic groups \({\text {Sp}}_m(2)\) for various m arising from certain representations of groups of Lie type.

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Acknowledgements

We would like to thank Günter Malle for helpful correspondence and the anonymous referee for a detailed report which aided in the exposition of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Bard College, Annandale-On-Hudson, NY, 12504, USA

    John Cullinan

  2. Department of Physics, Mathematics and Informatics, National Academy of Sciences of Belarus, 66 Prospekt Nezavisimosti, Minsk, Belarus

    Alexandre Zalesski

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  1. John Cullinan
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  2. Alexandre Zalesski
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Correspondence to John Cullinan.

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Dedicated to the memory of James Humphreys

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Cullinan, J., Zalesski, A. Unisingular representations in arithmetic and Lie theory. European Journal of Mathematics 7, 1645–1667 (2021). https://doi.org/10.1007/s40879-021-00496-3

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