Supercharacters for Normal Supercharacter Theory

TL;DR

This paper introduces a new class of supercharacter theories called normal supercharacter theories, constructed from normal subgroups, providing a tractable alternative to wild character theories for certain groups.

Contribution

It defines normal supercharacter theories from arbitrary normal subgroups, offers recursive formulas, and presents algorithms for their supercharacter tables, expanding the framework of supercharacter theory.

Findings

Normal supercharacter theories are integral.

Superclasses are determined by primitive central idempotents.

An algorithm for constructing supercharacter tables is provided.

Abstract

In order to find a tractable theory to substitute for the wild character theory of the group of $n\times n$ unipotent upper-triangular matrices over a finite field $\mathbb{F}_q$, Andr\'e and Yan introduced the notion of supercharacter theory. In this paper, we construct a supercharacter theory from an arbitrary set $S$ of normal subgroups of $G$. We call such supercharacter theory the normal supercharacter theory generated by $S$. It is shown that normal supercharacter theories are integral, and a recursive formula for supercharacters of the normal supercharacter theory is provided. The normal supercharacter theory provides many substitutions for wild character theories. Also, we indicate that the superclasses of the normal supercharacter theory generated by all normal subgroups of $G$ are given by certain values on the primitive central idempotents. We study the connection between the finest normal supercharacter theory and faithful irreducible characters. Moreover, an algorithm is presented to construct the supercharacter table of the finest normal supercharacter theory from the character table. Finally, We justify that normal supercharacter theories cannot be obtained by preceding supercharacter theory constructions.