Supercharacter theories and Schur rings

TL;DR

This paper explores the relationship between supercharacter theories and Schur rings, providing explicit connections and constructions that relate these algebraic structures in finite group theory.

Contribution

It explicitly links supercharacter theories with Schur rings and introduces constructions that correspond to known Schur ring methods.

Findings

Established explicit connections between supercharacter theories and Schur rings.

Provided constructions of supercharacter theories based on Schur ring techniques.

Enhanced understanding of algebraic structures in finite group representation theory.

Abstract

Diaconis and Isaacs have defined the supercharacter theories of a finite group to be certain approximations to the ordinary character theory of the group. We make explicit the connection between supercharacter theories and Schur rings, and we provide supercharacter theory constructions which correspond to Schur ring constructions of Leung and Man and of Tamaschke.