On the symmetric Gelfand pair $(\mathcal{H}_n\times \mathcal{H}_{n-1},diag (\mathcal{H}_{n-1}))$

TL;DR

This paper characterizes conjugacy classes in hyperoctahedral groups using marked bipartitions and proves that a specific pair forms a symmetric Gelfand pair with a multiplicity-free induced representation.

Contribution

It introduces a new parametrization of conjugacy classes in hyperoctahedral groups and establishes the symmetric Gelfand pair property for a specific group pair.

Findings

Conjugacy classes in $ ext{Hyperoctahedral}_n$ are indexed by marked bipartitions.

The pair $( ext{Hyperoctahedral}_n imes ext{Hyperoctahedral}_{n-1}, diag( ext{Hyperoctahedral}_{n-1}))$ is a symmetric Gelfand pair.

The induced representation from the diagonal subgroup is multiplicity free.

Abstract

We show that the $\mathcal{H}_{n-1}$-conjugacy classes of $\mathcal{H}_n,$ where $\mathcal{H}_n$ is the hyperoctahedral group on $2n$ elements, are indexed by marked bipartitions of $n.$ This will lead us to prove that $(\mathcal{H}_n\times \mathcal{H}_{n-1},diag (\mathcal{H}_{n-1}))$ is a symmetric Gelfand pair and that the induced representation $1_{diag (\mathcal{H}_{n-1})}^{\mathcal{H}_n\times \mathcal{H}_{n-1}}$ is multiplicity free.