In mathematics, and especially the discipline of representation theory, 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. Definition 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 If χ is irreducible, then its Frobenius–Schur indicator is 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. Much of the content below discusses the case of finite groups, but the general compact case is analogous. Real irreducible representations There are three types of irreducible real representations of a finite group on a real vector space V, as Schur's lemma implies that the endomorphism ring commuting with the group action is a real associative division algebra and by the Frobenius theorem can only be isomorphic to either the real numbers, or the complex numbers, or the quaternions. If the ring is the real numbers, then V⊗C is an irreducible complex representation with Schur indicator 1, also called a real representation. If the ring is the complex numbers, then V has two different conjugate complex structures, giving two irreducible complex representations with Schur indicator 0, sometimes called complex representations. If the ring is the quaternions, then choosing a subring of the quaternions isomorphic to the complex numbers makes V into an irreducible complex representation of G with Schur indicator −1, called a quaternionic representation. 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