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Let $\rho:G\to\GL_n(\C)$ be an irreducible complex group representation. Then $\rho$ is one of three types:

  1. Real if $\rho$ is conjugate to a representation $G\to \GL_n(\R)$
  2. Complex if some character value $\textrm{Tr}(\rho(g))$ is not contained in $\R$
  3. Quaternionic if the character values are all real but the representation is not conjugate to a real representation.

The type of the representation can be computed via its Frobenius-Schur indicator.

Knowl status:
  • Review status: reviewed
  • Last edited by Jennifer Paulhus on 2022-07-19 14:30:14
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