A supercharacter theory for involutive algebra groups (original) (raw)

Journal of Algebra, 2015

Abstract

ABSTRACT If mathscrJ\mathscr{J}mathscrJ is a finite-dimensional nilpotent algebra over a finite field Bbbk\BbbkBbbk, the algebra group P=1+mathscrJP = 1+\mathscr{J}P=1+mathscrJ admits a (standard) supercharacter theory as defined by Diaconis and Isaacs. If mathscrJ\mathscr{J}mathscrJ is endowed with an involution widehatvarsigma\widehat{\varsigma}widehatvarsigma, then widehatvarsigma\widehat{\varsigma}widehatvarsigma naturally defines a group automorphism of P=1+mathscrJP = 1+\mathscr{J}P=1+mathscrJ, and we may consider the fixed point subgroup CP(widehatvarsigma)=xinP:widehatvarsigma(x)=x−1C_{P}(\widehat{\varsigma}) = \{x\in P : \widehat{\varsigma}(x) = x^{-1}\}CP(widehatvarsigma)=xinP:widehatvarsigma(x)=x−1. Assuming that Bbbk\BbbkBbbk has odd characteristic ppp, we use the standard supercharacter theory for PPP to construct a supercharacter theory for CP(widehatvarsigma)C_{P}(\widehat{\varsigma})CP(widehatvarsigma). In particular, we obtain a supercharacter theory for the Sylow ppp-subgroups of the finite classical groups of Lie type, and thus extend in a uniform way the construction given by Andr\'e and Neto for the special case of the symplectic and orthogonal groups.

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