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Asymptotic Bounds for the Size of Hom(A,GL_n(q))

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JournalGlasgow Mathematical Journal
DateAccepted/In press - 12 Jul 2016
DateE-pub ahead of print - 14 Mar 2017
DatePublished (current) - 1 Jan 2018
Issue number1
Volume60
Number of pages11
Pages (from-to)51-61
Early online date14/03/17
Original languageEnglish

Abstract

Fix an arbitrary finite group A of order a, and let X(n,q) denote the set of homomorphisms from A to the finite general linear group GL_n(q). The size of X(n,q) is a polynomial in q. In this note it is shown that generically this polynomial has degree n^{2(1-a^{-1}) - \epsilon_r} and leading coefficient m_r, where \epsilon_r and m_r are constants depending only on r := n \mod a. We also present an algorithm for explicitly determining these constants.

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© Glasgow Mathematical Journal Trust 2017. This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details.

    Research areas

  • Finite Groups, Counting Homomorphisms, Representation Varieties

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