Finite Gelfand pairs and cracking points of the symmetric groups

Abstract

Let $\mathrm{\Gamma }$ be a finite group. Consider the wreath product $G_n:={\mathrm{\Gamma }}^n⋊S_n$ and the subgroup $K_n:={\mathrm{\Delta }}_n\times S_n\subseteq G_n$, where $S_n$ is the symmetric group and ${\mathrm{\Delta }}_n$ is the diagonal subgroup of ${\mathrm{\Gamma }}^n$. For certain values of $n$ (which depend on the group $\mathrm{\Gamma }$), the pair $\left(G_n,K_n\right)$ 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 $\mathrm{\Gamma }=S_k$, $k\ge 5$, $\left(G_n,K_n\right)$ is a Gelfand pair exactly when $n=1,2$.