Normal Supercharacter Theory

TL;DR

This paper introduces a new supercharacter theory construction for groups based on arbitrary sets of normal subgroups, expanding the existing methods by relating to central idempotent partitions.

Contribution

It presents a novel supercharacter theory framework from arbitrary normal subgroups, not derivable from automorphisms or single normal subgroups.

Findings

Constructs supercharacter theories from arbitrary normal subgroups.

Shows the new theories relate to partitions via central idempotents.

Demonstrates the theories cannot be obtained by previous methods.

Abstract

There are two main constructions of supercharacter theories for a group $ G $. The first, defined by Diaconis and Isaacs, comes from the action of a group $A$ via automorphisms on our given group $G$. The second, defined by Hendrickson, is combining a supercharacter theories of a normal subgroup $N$ of $G$ with a supercharacter theory of $G/N$. In this paper we construct a supercharacter theory from an arbitrary set of normal subgroups of $G$. We show that when consider the set of all normal subgroups of G, the corresponding supercharacter theory is related to a partition of $G$ given by certain values on the central idempotents. Also, we show the supercharacter theories that we construct can not be obtained via automorphisms or a single normal subgroup.