# Frobenius-Schur indicator

From Groupprops

## Contents

## Definition

Let be a finite group and a character or virtual character of . The **Frobenius-Schur indicator** of is the value:

Equivalently:

where varies over a collection of conjugacy class representatives and denotes the centralizer of in .

Equivalently, is the inner product of and the indicator character.

## Particular cases

### Examples where the Frobenius-Schur indicator is -1

Representation | Computation of Frobenius-Schur indicator (section) | Group | Information on linear representation theory |
---|---|---|---|

faithful irreducible representation of quaternion group | faithful irreducible representation of quaternion group#Frobenius-Schur indicator | quaternion group | linear representation theory of quaternion group |

quaternionic representation of special linear group:SL(2,3) | quaternionic representation of special linear group:SL(2,3)#Frobenius-Schur indicator | special linear group:SL(2,3) | linear representation theory of special linear group:SL(2,3) |

## Facts

### For irreducible characters

`Further information: Indicator theorem`

If is an the character of an irreducible representation, then is either 0, +1, or -1:

- if and only if is the character of a representation over
- if and only if is a real-valued character, but cannot be realized as the character of a real representation
- if and only if some value of is non-real