Level-Rank Dualities for Finite Reductive Groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916890205487104 |
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| author | Trinh, Minh-Tâm Quang Xue, Ting |
| author_facet | Trinh, Minh-Tâm Quang Xue, Ting |
| contents | This is an extended abstract of our work "Level-Rank Dualities from $Φ$-Cuspidal Pairs..." We present evidence for a family of surprising coincidences within the representation theory of a finite reductive group $G$: more precisely, dualities between blocks of cyclotomic Hecke algebras attached by Broué-Malle to $Φ$-cuspidal pairs of $G$, where the Hecke parameters are specialized not to the order of the underlying finite field, but to roots of unity. For the groups $G = \mathrm{GL}_n(\mathbf{F}_q)$, these coincidences can be expressed very concretely in terms of the combinatorics of partitions, and the whole story recovers an avatar of the level-rank duality studied by Frenkel, Uglov, Chuang-Miyachi, and others. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_07051 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Level-Rank Dualities for Finite Reductive Groups Trinh, Minh-Tâm Quang Xue, Ting Representation Theory Combinatorics This is an extended abstract of our work "Level-Rank Dualities from $Φ$-Cuspidal Pairs..." We present evidence for a family of surprising coincidences within the representation theory of a finite reductive group $G$: more precisely, dualities between blocks of cyclotomic Hecke algebras attached by Broué-Malle to $Φ$-cuspidal pairs of $G$, where the Hecke parameters are specialized not to the order of the underlying finite field, but to roots of unity. For the groups $G = \mathrm{GL}_n(\mathbf{F}_q)$, these coincidences can be expressed very concretely in terms of the combinatorics of partitions, and the whole story recovers an avatar of the level-rank duality studied by Frenkel, Uglov, Chuang-Miyachi, and others. |
| title | Level-Rank Dualities for Finite Reductive Groups |
| topic | Representation Theory Combinatorics |
| url | https://arxiv.org/abs/2508.07051 |