Indicator theorem

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Statement

The following information about an irreducible linear representation over complex numbers can be garnered from the Frobenius-Schur indicator of its character \chi (denoted v(\chi)) (viz, its inner product with the indicator character):

  • v(\chi) = 0 if and only if \chi is not real-valued
  • v(\chi) = 1 if and only if \chi is the character of a real representation
  • v(\chi)=-1 if and only if \chi is real-valued but does not arise as the character of a real representation

Proof

Let \rho:G \to GL(V) be the linear representation giving the character \chi. Denote by Sym^2(\rho) the corresponding representation on Sym^2(V) and by Alt^2(\rho) the corresponding representation on Alt^2(V). Let \chi_A be the character of Alt^2(\rho).

Then, by some elementary computations:

2\chi_A(g) = \chi(g)^2 - \chi(g^2)

From this it follows that:

v(\chi) = (1_G, \chi^2 - 2\chi_A) = (1_G,\chi^2) - 2(1_G,\chi_A)

Now since Alt^2(\rho) is a direct summand of \rho \otimes \rho, we must have 1_G,\chi_A) \le (1_G,\chi^2). But if \chi is not real-valued, then (1_G,\chi^2) = 0 so v(\chi) = 0 and if \chi is real-valued, then (1_G,\chi^2) = 1. Now two cases arise:

  • (1_G,\chi_A) = 1 and hence v(\chi) = -1. This happens only if \chi is not the character of a real representation.
  • (1_G,\chi_A) = 0 and hence v(\chi) = 1. This happens only if \chi 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.