A Supercharacter Analogue for Normality

TL;DR

This paper extends supercharacter theory to algebra subgroups called supernormal, classifies such subgroups in unipotent upper triangular matrices, and develops supercharacter analogues of classical theorems like Clifford's and Mackey's.

Contribution

It introduces the concept of supernormal subgroups, classifies them in certain algebra groups, and adapts key representation theorems to the supercharacter framework.

Findings

Supernormal subgroups are closed under products.

All normal pattern subgroups are supernormal.

Supercharacter restrictions decompose similarly to classical characters.

Abstract

Diaconis and Isaacs define a supercharacter theory for algebra groups over a finite field by constructing certain unions of conjugacy classes called superclasses and certain reducible characters called supercharacters. This work investigates the properties of algebra subgroups $H\subset G$ which are unions of some set of the superclasses of $G$; we call such subgroups supernormal. After giving a few useful equivalent formulations of this definition, we show that products of supernormal subgroups are supernormal and that all normal pattern subgroups are supernormal. We then classify the set of supernormal subgroups of $U_n(q)$, the group of unipotent upper triangular matrices over the finite field $\FF_q$, and provide a formula for the number of such subgroups when $q$ is prime. Following this, we give supercharacter analogues for Clifford's theorem and Mackey's "method of little groups." Specifically, we show that a supercharacter restricted to a supernormal subgroup decomposes as a sum of supercharacters with the same degree and multiplicity. We then describe how the supercharacters of an algebra group of the form $U_\fkn = U_\fkh \ltimes U_\fka$, where $U_\fka$ is supernormal and $\fka^2=0$, are parametrized by $U_\fkh$-orbits of the supercharacters of $U_\fka$ and the supercharacters of the stabilizer subgroups of these orbits.