Editing Frobenius–Schur indicator (section)

==Definition==

If a finite-dimensional continuous representation of a compact group ''G'' has [[character theory|character]] &chi; its '''Frobenius–Schur indicator''' is defined to be

:<math>\int_{g\in G}\chi(g^2)\,d\mu</math>

for [[Haar measure]] μ with μ(''G'') = 1. When ''G'' is [[finite group|finite]] it is given by

:<math>{1\over |G|}\sum_{g\in G}\chi(g^2).</math>

If &chi; is a complex irreducible representation, then its Frobenius–Schur indicator is 1, 0, or −1.  It provides a criterion for deciding whether a real irreducible representation of ''G'' is real, complex or [[quaternion]]ic, in a specific sense defined below. Much of the content below discusses the case of finite groups, but the general compact case is analogous.