Wreath Generalization of Littlewood Reciprocity
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915467456675840 |
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| author | Weising, Milo Bechtloff |
| author_facet | Weising, Milo Bechtloff |
| contents | Given any $m$-dimensional complex representation $η$ of a finite group $G$ and any highest weight representation $V^λ$ of $\mathrm{GL}_{nm}(\mathbb{C})$ we may define an action of $G^n \rtimes \mathfrak{S}_n$ on $V^λ$ using the embedding $\mathrm{GL}_{m}(\mathbb{C})^n \rtimes \mathfrak{S}_n \leq \mathrm{GL}_{nm}(\mathbb{C})$ and $η: G \rightarrow \mathrm{GL}_m(\mathbb{C})$. We derive a branching rule for the multiplicities of irreducible $G^n \rtimes \mathfrak{S}_n$ representations in $V^λ.$ The formula generalizes Littlewood's reciprocity rule for branching between $\mathrm{GL}_n(\mathbb{C})$ and the symmetric group of permutation matrices $\mathfrak{S}_n \leq \mathrm{GL}_n(\mathbb{C}).$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_07727 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Wreath Generalization of Littlewood Reciprocity Weising, Milo Bechtloff Combinatorics Representation Theory Given any $m$-dimensional complex representation $η$ of a finite group $G$ and any highest weight representation $V^λ$ of $\mathrm{GL}_{nm}(\mathbb{C})$ we may define an action of $G^n \rtimes \mathfrak{S}_n$ on $V^λ$ using the embedding $\mathrm{GL}_{m}(\mathbb{C})^n \rtimes \mathfrak{S}_n \leq \mathrm{GL}_{nm}(\mathbb{C})$ and $η: G \rightarrow \mathrm{GL}_m(\mathbb{C})$. We derive a branching rule for the multiplicities of irreducible $G^n \rtimes \mathfrak{S}_n$ representations in $V^λ.$ The formula generalizes Littlewood's reciprocity rule for branching between $\mathrm{GL}_n(\mathbb{C})$ and the symmetric group of permutation matrices $\mathfrak{S}_n \leq \mathrm{GL}_n(\mathbb{C}).$ |
| title | Wreath Generalization of Littlewood Reciprocity |
| topic | Combinatorics Representation Theory |
| url | https://arxiv.org/abs/2506.07727 |