A FAMILY OF FINITE GELFAND PAIRS ASSOCIATED WITH WREATH PRODUCTS

Consider the wreath product G(n) = Gamma(n) (sic) S-n of a finite group Gamma with the symmetric group S-n. Let Delta(n) denote the diagonal in Gamma(n). Then K-n = Delta(n) x S-n forms a subgroup of G(n). In case Gamma is abelian, (G(n), K-n) forms a Gelfand pair with relevance to the study of parking functions. For Gamma non-abelian it was suggested by Kursat Aker and Mahir Bilen Can that (G(n), K-n) must fail to be a Gelfand pair for n sufficiently large. We prove that this is indeed the case: for F non-abelian there is some integer 2 < N(Gamma) <= vertical bar Gamma vertical bar for which (K-n, G(n)) is a Gelfand pair for all n < N(Gamma) but (K-n, G(n)) fails to be a Gelfand pair for all n >= N(Gamma). For dihedral groups Gamma = D-p, with p an odd prime we prove that N(Gamma) = 6.