On the Springer correspondence for wreath products
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915968376111104 |
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| author | Hsu, You-Hung Lai, Chun-Ju |
| author_facet | Hsu, You-Hung Lai, Chun-Ju |
| contents | We establish a Bruhat decomposition indexed by the wreath product $Σ_m\wr Σ_d$ between two symmetric groups -- note that $Σ_m\wr Σ_d$ is not a Coxeter group in general. We show that such a decomposition affords a geometric variant in terms of the Bialynicki-Birula decomposition for varieties with $\mathbb{C}^*$-actions. Next, we construct a Steinberg variety whose top Borel-Moore homology realizes the group algebra $\mathbb{Q}[Σ_m\wr Σ_d]$ as a proper subalgebra. Such a geometric realization leads to a Springer-type correspondence which identifies the irreducible representations of $Σ_m\wr Σ_d$ with isotypic components of certain unconventional Springer fibers using type A geometry. In other words, we obtain a geometric counterpart of the (algebraic) Clifford theory, for the first time. Consequently, we obtain a new Springer correspondence of Weyl groups of type B/C/D using essentially type A geometry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_02846 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Springer correspondence for wreath products Hsu, You-Hung Lai, Chun-Ju Representation Theory Algebraic Geometry We establish a Bruhat decomposition indexed by the wreath product $Σ_m\wr Σ_d$ between two symmetric groups -- note that $Σ_m\wr Σ_d$ is not a Coxeter group in general. We show that such a decomposition affords a geometric variant in terms of the Bialynicki-Birula decomposition for varieties with $\mathbb{C}^*$-actions. Next, we construct a Steinberg variety whose top Borel-Moore homology realizes the group algebra $\mathbb{Q}[Σ_m\wr Σ_d]$ as a proper subalgebra. Such a geometric realization leads to a Springer-type correspondence which identifies the irreducible representations of $Σ_m\wr Σ_d$ with isotypic components of certain unconventional Springer fibers using type A geometry. In other words, we obtain a geometric counterpart of the (algebraic) Clifford theory, for the first time. Consequently, we obtain a new Springer correspondence of Weyl groups of type B/C/D using essentially type A geometry. |
| title | On the Springer correspondence for wreath products |
| topic | Representation Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2404.02846 |