FROBENIUS–SCHUR INDICATORS FOR PROJECTIVE CHARACTERS WITH APPLICATIONS

Abstract Let $\alpha $ be a complex valued $2$ -cocycle of finite order of a finite group $G.$ The n th Frobenius–Schur indicator of an irreducible $\alpha $ -character of G is defined and its properties are investigated. The indicator is interpreted in general for $n =2$ and it is shown that it can be used to determine whether an irreducible $\alpha $ -character is real-valued under the assumption that the order of $\alpha $ and its cohomology class are both $2$ . A formula, involving the real $\alpha $ -regular conjugacy classes of $G,$ is found to count the number of real-valued irreducible $\alpha $ -characters of G under the additional assumption that these characters are class functions.