Real representation

A criterion (for compact groups G) for reality of irreducible representations in terms of character theory is based on the Frobenius-Schur indicator defined by :\int_{g\in G}\chi(g^2)\,d\mu where χ is the character of the representation and μ is the Haar measure with μ(G) = 1. For a finite group, this is given by :{1\over |G|}\sum_{g\in G}\chi(g^2). The indicator may take the values 1, 0 or −1. If the indicator is 1, then the representation is real. If the indicator is zero, the representation is complex (hermitian), and if the indicator is −1, the representation is quaternionic. ==Examples==

All representation of the symmetric groups are real (and in fact rational), since we can build a complete set of irreducible representations using Young tableaux. All representations of the rotation groups on odd-dimensional spaces are real, since they all appear as subrepresentations of tensor products of copies of the fundamental representation, which is real. Further examples of real representations are the spinor representations of the spin groups in 8k−1, 8k, and 8k+1 dimensions for k = 1, 2, 3 ... This periodicity modulo 8 is known in mathematics not only in the theory of Clifford algebras, but also in algebraic topology, in KO-theory; see spin representation and Bott periodicity. ==Notes==