A supercharacter theory for involutive algebra groups

If \(\mathscr{J}\) is a finite-dimensional nilpotent algebra over a finite field \(\Bbbk\), the algebra group \(P = 1+\mathscr{J}\) admits a (standard) supercharacter theory as defined by Diaconis and Isaacs. If \(\mathscr{J}\) is endowed with an involution \(\widehat{\varsigma}\), then \(\widehat{\...

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Bibliographic Details
Published inarXiv.org
Main Authors André, Carlos A M, Freitas, Pedro J, Neto, Ana Margarida
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 05.02.2015
Subjects
Algebra
Automorphisms
Fields (mathematics)
Group theory
Subgroups
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Summary:If \(\mathscr{J}\) is a finite-dimensional nilpotent algebra over a finite field \(\Bbbk\), the algebra group \(P = 1+\mathscr{J}\) admits a (standard) supercharacter theory as defined by Diaconis and Isaacs. If \(\mathscr{J}\) is endowed with an involution \(\widehat{\varsigma}\), then \(\widehat{\varsigma}\) naturally defines a group automorphism of \(P = 1+\mathscr{J}\), and we may consider the fixed point subgroup \(C_{P}(\widehat{\varsigma}) = \{x\in P : \widehat{\varsigma}(x) = x^{-1}\}\). Assuming that \(\Bbbk\) has odd characteristic \(p\), we use the standard supercharacter theory for \(P\) to construct a supercharacter theory for \(C_{P}(\widehat{\varsigma})\). In particular, we obtain a supercharacter theory for the Sylow \(p\)-subgroups of the finite classical groups of Lie type, and thus extend in a uniform way the construction given by André and Neto for the special case of the symplectic and orthogonal groups.
ISSN:2331-8422