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:

$\mathrm{i}\mathrm{n}\mathrm{d}(\alpha )={\sum }_{x\in G}\alpha (x^2)/\left|G\right|$

Equivalently:

$\mathrm{i}\mathrm{n}\mathrm{d}(\alpha )=\sum \alpha (x^2)/|C_G(x)|$

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

Equivalently, $\mathrm{i}\mathrm{n}\mathrm{d}(\alpha )$ is the inner product of $\alpha$ and the indicator character.

Particular cases

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)

Facts

For irreducible characters

Further information: Indicator theorem

If $\chi$ is an the character of an irreducible representation, then $\mathrm{i}\mathrm{n}\mathrm{d}(\chi )$ is either 0, +1, or -1:

• $\mathrm{i}\mathrm{n}\mathrm{d}(\chi )=+1$ if and only if $\chi$ is the character of a representation over $\mathbb{R}$
• $\mathrm{i}\mathrm{n}\mathrm{d}(\chi )=-1$ if and only if $\chi$ is a real-valued character, but cannot be realized as the character of a real representation
• $\mathrm{i}\mathrm{n}\mathrm{d}(\chi )=0$ if and only if some value of $\chi$ is non-real