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.
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)|
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