Generalized characters of the generalized symmetric group

TL;DR

This paper establishes that a certain wreath product group forms a symmetric Gelfand pair, introduces generalized characters based on zonal spherical functions, and provides a Murnaghan-Nakayama rule for these characters, extending previous work on symmetric groups.

Contribution

It proves the symmetric Gelfand pair property for wreath products of cyclic groups with symmetric groups and defines generalized characters with properties similar to classical characters.

Findings

Proves the symmetric Gelfand pair property for the wreath product groups.

Defines generalized characters using zonal spherical functions.

Provides a Murnaghan-Nakayama rule for the generalized characters.

Abstract

We prove that $(\mathbb{Z}_k \wr \mathcal{S}_n \times \mathbb{Z}_k \wr \mathcal{S}_{n-1}, \text{diag} (\mathbb{Z}_k \wr \mathcal{S}_{n-1}) )$ is a symmetric Gelfand pair, where $\mathbb{Z}_k \wr \mathcal{S}_n$ is the wreath product of the cyclic group $\mathbb{Z}_k$ with the symmetric group $\mathcal{S}_n.$ The proof is based on the study of the $\mathbb{Z}_k \wr \mathcal{S}_{n-1}$-conjugacy classes of $\mathbb{Z}_k \wr \mathcal{S}_n.$ We define the generalized characters of $\mathbb{Z}_k \wr \mathcal{S}_n$ using the zonal spherical functions of $(\mathbb{Z}_k \wr \mathcal{S}_n \times \mathbb{Z}_k \wr \mathcal{S}_{n-1}, \text{diag} (\mathbb{Z}_k \wr \mathcal{S}_{n-1}) ).$ We show that these generalized characters have properties similar to usual characters. A Murnaghan-Nakayama rule for the generalized characters of the hyperoctahedral group is presented. The generalized characters of the symmetric group were first studied by Strahov in [7].