FINITE GELFAND PAIRS AND CRACKING POINTS OF THE SYMMETRIC GROUPS

Let Gamma be a finite group. Consider the wreath product G(n) := Gamma(n) (sic) S-n and the subgroup K-n := Delta(n) x S-n subset of G(n), where S-n is the symmetric group and Delta(n) is the diagonal subgroup of Gamma(n). For certain values of n (which depend on the group Gamma), the pair (G(n), K-n) is a Gelfand pair. It is not known for all finite groups which values of n result in Gelfand pairs. Building off the work of Benson-Ratcliff [4], we obtain a result which simplifies the computation of multiplicities of irreducible representations in certain tensor product representations, then apply this result to show that for Gamma = S-k, k >= 5, (G(n), K-n) is a Gelfand pair exactly when n = 1, 2.