Wreath Generalization of Littlewood Reciprocity

TL;DR

This paper generalizes Littlewood's reciprocity rule by deriving a branching formula for representations of a wreath product of a finite group with the symmetric group, acting on highest weight representations of general linear groups.

Contribution

It introduces a wreath generalization of Littlewood reciprocity, providing a new branching rule for irreducible representations involving finite groups and symmetric groups.

Findings

Derived a new branching rule for $G^n times rak{S}_n$ representations

Generalized Littlewood's reciprocity to wreath product settings

Provided explicit multiplicity formulas for irreducible components

Abstract

Given any $m$-dimensional complex representation $\eta$ of a finite group $G$ and any highest weight representation $V^{\lambda}$ of $\mathrm{GL}_{nm}(\mathbb{C})$ we may define an action of $G^n \rtimes \mathfrak{S}_n$ on $V^{\lambda}$ using the embedding $\mathrm{GL}_{m}(\mathbb{C})^n \rtimes \mathfrak{S}_n \leq \mathrm{GL}_{nm}(\mathbb{C})$ and $\eta: G \rightarrow \mathrm{GL}_m(\mathbb{C})$. We derive a branching rule for the multiplicities of irreducible $G^n \rtimes \mathfrak{S}_n$ representations in $V^{\lambda}.$ The formula generalizes Littlewood's reciprocity rule for branching between $\mathrm{GL}_n(\mathbb{C})$ and the symmetric group of permutation matrices $\mathfrak{S}_n \leq \mathrm{GL}_n(\mathbb{C}).$