Frobenius-Schur indicator

Definition
Let $$G$$ be a finite group and $$\alpha$$ a character or virtual character of $$G$$. The Frobenius-Schur indicator of $$\alpha$$ is the value:

$$\operatorname{ind}(\alpha) = \sum_{x \in G} \alpha(x^2)/|G|$$

Equivalently:

$$\operatorname{ind}(\alpha) = \sum \alpha(x^2)/|C_G(x)|$$

where $$x$$ varies over a collection of conjugacy class representatives and $$C_G(x)$$ denotes the centralizer of $$x$$ in $$G$$.

Equivalently, $$\operatorname{ind}(\alpha)$$ is the inner product of $$\alpha$$ and the indicator character.

For irreducible characters
If $$\chi$$ is an the character of an irreducible representation, then $$\operatorname{ind}(\chi)$$ is either 0, +1, or -1:


 * $$\operatorname{ind}(\chi) = +1$$ if and only if $$\chi$$ is the character of a representation over $$\R$$
 * $$\operatorname{ind}(\chi) = -1$$ if and only if $$\chi$$ is a real-valued character, but cannot be realized as the character of a real representation
 * $$\operatorname{ind}(\chi) = 0$$ if and only if some value of $$\chi$$ is non-real