Real Representation - Frobenius-Schur Indicator

A criterion (for compact groups G) for reality of irreducible representations in terms of character theory is based on the Frobenius-Schur indicator defined by

where χ is the character of the representation and μ is the Haar measure with μ(G) = 1. For a finite group, this is given by

The indicator may take the values 1, 0 or −1. If the indicator is 1, then the representation is real. If the indicator is zero, the representation is complex (hermitian), and if the indicator is −1, the representation is quaternionic.