FROBENIUS-SCHUR INDICATORS FOR PROJECTIVE CHARACTERS WITH APPLICATIONS

Let alpha be a complex valued 2 -cocycle of finite order of a finite group G. The nth 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.