Wreath Generalization of Littlewood Reciprocity

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 $G^n⋊{\mathfrak{S}}_n$ on $V^{\lambda }$ using the embedding ${\mathrm{GL}}_m(\mathbb{C}{)}^n⋊{\mathfrak{S}}_n\le {\mathrm{GL}}_{nm}(\mathbb{C})$ and $\eta :G\to {\mathrm{GL}}_m(\mathbb{C})$. We derive a branching rule for the multiplicities of irreducible $G^n⋊{\mathfrak{S}}_n$ 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 ${\mathfrak{S}}_n\le {\mathrm{GL}}_n(\mathbb{C}).$

10 pages; extends result from previous version to all finite groups