# Conjugacy class-representation bijection

## Contents

## Definition

Given a finite group and a field that is sufficiently large for the finite group, a **conjugacy class-representation bijection** for the finite group and the field is a bijection between the set of conjugacy classes in the finite group and the set of representations (upto equivalence) of the finite group over the field.

## Properties

### Self-adjointness

The bijection is termed **self-adjoint** if it takes inverse elements to contragredient representations: Under the bijection, two conjugacy classes that are inverse of each other, get mapped to representations that are contragredient.

### Characteristic class-representation bijection

A conjugacy class-representation bijection is said to be characteristic if it commutes with any automorphism. Here, we use the fact that the automorphism group acts both on the set of conjugacy classes and on the set of equivalence classes of representations.

Note that in a complete group, any class-representation bijection is characteristic.