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 $widehat{varsigma}$ naturally defines a group automorphism of $P = 1+mathscr{J}$, and we may consider the fixed point subgroup $C_{P}(widehat{varsigma}) = {xin P : widehat{varsigma}(x) = x^{-1}}$. 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 Andre and Neto for the special case of the symplectic and orthogonal groups.

A supercharacter theory for the Sylow p-subgroups of the finite symplectic and orthogonal groups

We define the superclasses for a classical finite unipotent group $U$ of type $B_{n}(q)$, $C_{n}(q)$, or $D_{n}(q)$, and show that, together with the supercharacters defined in a previous paper, they form a supercharacter theory. In particular, we prove that the supercharacters take a constant value on each superclass, and evaluate this value. As a consequence, we obtain a factorization of any superclass as a product of elementary superclasses. In addition, we also define the space of superclass functions, and prove that it is spanned by the supercharacters. As as consequence, we (re)obtain the decomposition of the regular character as an orthogonal linear combination of supercharacters. Finally, we define the supercharacter table of $U$, and prove various orthogonality relations for supercharacters (similar to the well-known orthogonality relations for irreducible characters).

Supercharacters of the Sylow p-subgroups of the finite symplectic and orthogonal groups

We define and study supercharacters of the classical finite unipotent groups of symplectic and orthogonal types (over any finite field of odd characteristic). We show how supercharacters for groups of those types can be obtained by restricting the supercharacter theory of the finite unitriangular group, and prove that supercharacters are orthogonal and provide a partition of the set of all irreducible characters. We also describe all irreducible characters of maximum degree in terms of the root system, and show how they can be obtained as constituents of particular supercharacters.