Wreath Generalization of Littlewood Reciprocity

Milo Bechtloff Weising

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arXiv · 2506.07727

Wreath Generalization of Littlewood Reciprocity

Milo Bechtloff Weising

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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 $\mathfrak{S}_n \leq \mathrm{GL}_n(\mathbb{C}).$

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Milo Bechtloff Weising. 2025-06-09. Wreath Generalization of Littlewood Reciprocity. https://arxiv.org/abs/2506.07727

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