Frobenius-Schur indicator

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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.

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 \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