Finite Gelfand pairs and cracking points of the symmetric groups

Abstract

Let $\Gamma$ be a finite group. Consider the wreath product $G_n:={\Gamma }^n⋊S_n$ and the subgroup $K_n:={\Delta }_n\times S_n\subseteq G_n$, 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, 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 $\Gamma =S_k,k\ge 5$, $(G_n,K_n)$ is a Gelfand pair exactly when $n=1,2$.

6 pages; preliminary version, comments welcome