Finding Supercharacter Theories on Character Tables

TL;DR

This paper presents a method and algorithm for efficiently finding all supercharacter theories of a finite group using its character table, highlighting the duality between superclasses and supercharacters.

Contribution

It introduces a systematic approach and an algorithm to identify supercharacter theories from character tables, including computations for certain nonabelian simple groups.

Findings

Algorithm successfully finds all supercharacter theories for given groups.

Computed the number of supercharacter theories for some nonabelian simple groups.

Highlights the duality between superclasses and supercharacters.

Abstract

We describe an easy way how to find supercharacter theories for a finite group, if its character table is known. Namely, we show how an arbitrary partition of the conjugacy classes or of the irreducible characters can be refined to the coarsest partition that belongs to a supercharacter theory. Our constructions emphasize the duality between superclasses and supercharacters. An algorithm is presented to find all supercharacter theories on a given character table. The algorithm is used to compute the number of supercharacter theories for some nonabelian simple groups with up to 26 conjugacy classes.