Finite Gelfand pairs and cracking points of the symmetric groups

TL;DR

This paper characterizes when certain wreath product group pairs form Gelfand pairs, showing for symmetric groups with k ≥ 5, the pairs are Gelfand only at n=1,2, simplifying multiplicity calculations.

Contribution

It provides a new criterion for identifying Gelfand pairs in wreath products and applies it to symmetric groups, extending previous work.

Findings

Gelfand pairs occur only at n=1,2 for Γ=S_k, k≥5

New method simplifies multiplicity computations in tensor products

Extends understanding of Gelfand pairs in wreath product groups

Abstract

Let $\Gamma$ be a finite group. Consider the wreath product $G_n := \Gamma^n \rtimes S_n$ 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 $\Gamma = S_k, \ k \geq 5$, $(G_n,K_n)$ is a Gelfand pair exactly when $n = 1,2$.