Finite Gelfand pairs and cracking points of the symmetric groups

2019/08/29 by Pearson, Faith, Romanov, Anna, Soller, Dylan
#20C15 #20C30 #20E22 #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1908.11045

Abstract

Let Γ be a finite group. Consider the wreath product G_n := Γⁿ \rtimes S_n and the subgroup K_n := Δ_n × S_n⊆ G_n, where S_n is the symmetric group and Δ_n is the diagonal subgroup of Γⁿ. For certain values of n (which depend on the group Γ), 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 Γ= S_k, k ≥ 5, (G_n,K_n) is a Gelfand pair exactly when n = 1,2.

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