Gelfand pairs involving the wreath product of finite abelian groups with symmetric groups

Abstract

It is well known that the pair $({\mathcal{S}}_n,{\mathcal{S}}_{n-1})$ is a Gelfand pair where ${\mathcal{S}}_n$ is the symmetric group on $n$ elements. In this paper, we prove that if $G$ is a finite group then $(G\wr {\mathcal{S}}_n,G\wr {\mathcal{S}}_{n-1}),$ where $G\wr {\mathcal{S}}_n$ is the wreath product of $G$ by ${\mathcal{S}}_n,$ is a Gelfand pair if and only if $G$ is abelian.

6 pages