Wreath Generalization of Littlewood Reciprocity

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