The Kronecker power of a permutation

Let Sn denote the full symmetric permutation group of degree n. For each a E e, let (J (a) be the corresponding permutation matrix, i.e., Q(a) = (0 iUU»)' If e is any subgroup of S n, then Q is a faithful representation of e whose c haracter, e, counts the number of fixed points . In trus note, we in vestigate a red uction of fr, the character of the rth Kronecker power of (J . The reduction of the Kronecker (or inner) product of two irred ucible representations is called a Clebsch-Gordon series. When e = S'" the proble m of obtaining a Clebsch-Gordon series has been solved (see, e.g., [3],1 [4] or [7]). However, the solution does not eas ily lead to explicit formulas for the red uction of higher Kronecker powers of re presentations. When 1 :::; r :::; n , the problem naturally arises in connection with a ce rtain class of matrix function s: Let iI. be an irreducible character of e. If A = (au) is an n-square co mplex matrix, Je t