Frobenius-Schur indicator

In mathematics the Schur indicator, named after Issai Schur, or Frobenius–Schur indicator describes what invariant bilinear forms a given irreducible representation of a compact group on a complex vector space has. It can be used to classify the irreducible representations of compact groups on real vector spaces.

If a finite-dimensional continuous complex representation of a compact group G has character χ its Frobenius–Schur indicator is defined to be

for Haar measure μ with μ(G) = 1. When G is finite it is given by

The Frobenius–Schur indicator is always 1, 0, or -1. It provides a criterion for deciding whether an irreducible representation of G is real, complex or quaternionic, in a specific sense defined below. Below we discuss the case of finite groups, but the general compact case is completely analogous.

There are three types of irreducible real representations of a finite group on a real vector space V, as the endomorphism ring commuting with the group action can be isomorphic to either the real numbers, or the complex numbers, or the quaternions.

Moreover every irreducible representation on a complex vector space can be constructed from a unique irreducible representation on a real vector space in one of the three ways above. So knowing the irreducible representations on complex spaces and their Schur indicators allows one to read off the irreducible representations on real spaces.

Real representations can be complexified to get a complex representation of the same dimension and complex representations can be converted into a real representation of twice the dimension by treating the real and imaginary components separately. Also, since all finite dimensional complex representations can be turned into a unitary representation, for unitary representations the dual representation is also a (complex) conjugate representation because the Hilbert space norm gives an antilinear bijective map from the representation to its dual representation.

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