G(l,k,d)-modules via groupoids

TL;DR

This paper introduces a novel combinatorial approach to the complex representation theory of wreath products involving finite abelian groups, utilizing groupoids and Schur-Weyl duality to classify modules and extend to complex reflection groups.

Contribution

It constructs a groupoid-based framework for wreath product representations, providing new classification methods and extending to complex reflection groups.

Findings

Constructed a groupoid with endomorphism algebras as symmetric groups

Proved the groupoid algebra is isomorphic to the wreath product group algebra

Derived a Gelfand model for wreath products

Abstract

In this note we describe a seemingly new approach to the complex representation theory of the wreath product $G\wr S_d$ where $G$ is a finite abelian group. The approach is motivated by an appropriate version of Schur-Weyl duality. We construct a combinatorially defined groupoid in which all endomorphism algebras are direct products of symmetric groups and prove that the groupoid algebra is isomorphic to the group algebra of $G\wr S_d$. This directly implies a classification of simple modules. As an application, we get a Gelfand model for $G\wr S_d$ from the classical involutive Gelfand model for the symmetric group. We describe the Schur-Weyl duality which motivates our approach and relate it to various Schur-Weyl dualities in the literature. Finally, we discuss an extension of these methods to all complex reflection groups of type $G(\ell,k,d)$.