The following information about an irreducible linear representation over complex numbers can be garnered from the Frobenius-Schur indicator of its character (denoted ) (viz, its inner product with the indicator character):
- if and only if is not real-valued
- if and only if is the character of a real representation
- if and only if is real-valued but does not arise as the character of a real representation
Let be the linear representation giving the character . Denote by the corresponding representation on and by the corresponding representation on . Let be the character of .
Then, by some elementary computations:
From this it follows that:
Now since is a direct summand of , we must have . But if is not real-valued, then so and if is real-valued, then . Now two cases arise:
- and hence . This happens only if is not the character of a real representation.
- and hence . This happens only if is the character of a real representation.
The idea behind proving the latter distinction is to relate this with the existence of a group-invariant symmetric bilinear form.