Generalized Involution Models for Wreath Products

TL;DR

This paper extends the concept of involution models to wreath products of finite groups, constructs Gelfand models for specific wreath products, and confirms conjectures about their properties and irreducible decompositions.

Contribution

It proves that wreath products of groups with involution models also have involution models and constructs explicit Gelfand models for certain wreath products.

Findings

Wreath products of groups with involution models inherit involution models.

Constructed Gelfand models for $A \,\wr\, S_n$ with $A$ abelian.

Provided an alternative proof for a Gelfand model of $\\ZZ_r \,\wr\, S_n$.

Abstract

We prove that if a finite group $H$ has a generalized involution model, as defined by Bump and Ginzburg, then the wreath product $H \wr S_n$ also has a generalized involution model. This extends the work of Baddeley concerning involution models for wreath products. As an application, we construct a Gelfand model for wreath products of the form $A \wr S_n$ with $A$ abelian, and give an alternate proof of a recent result due to Adin, Postnikov, and Roichman describing a particularly elegant Gelfand model for the wreath product $\ZZ_r \wr S_n$. We conclude by discussing some notable properties of this representation and its decomposition into irreducible constituents, proving a conjecture of Adin, Roichman, and Postnikov's.