A supercharacter theory for involutive algebra groups

TL;DR

This paper develops a supercharacter theory for involutive algebra groups, specifically fixed point subgroups under involution, extending previous work to classical groups of Lie type and their Sylow p-subgroups.

Contribution

It introduces a supercharacter theory for fixed point subgroups of algebra groups under involution, generalizing prior constructions for symplectic and orthogonal groups.

Findings

Constructed supercharacter theory for fixed point subgroups under involution.

Extended supercharacter theory to Sylow p-subgroups of classical groups.

Provided a uniform approach for classical groups of Lie type.

Abstract

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 Andr\'e and Neto for the special case of the symplectic and orthogonal groups.