On finite group scheme-theoretical categories, I
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866910293750185984 |
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| author | Gelaki, Shlomo Sanmarco, Guillermo |
| author_facet | Gelaki, Shlomo Sanmarco, Guillermo |
| contents | Let $\mathscr {C}(G,H,ψ)$ be a finite group scheme-theoretical category over an algebraically closed field of characteristic $p\ge 0$ as introduced by the first author. For any indecomposable exact module category over $\mathscr {C}(G,H,ψ)$, we classify its simple objects and provide an expression for their projective covers, in terms of double cosets and projective representations of certain closed subgroup schemes, which upgrades a result by Ostrik for group-theoretical fusion categories. As a byproduct, we describe the simples and indecomposable projectives of $\mathscr {C}(G,H,ψ)$, and parametrize the Brauer-Piccard group of ${\rm Coh}(G)$ for any finite connected group scheme $G$. Finally, we apply our results to describe the blocks of the center of ${\rm Coh}(G)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_10205 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On finite group scheme-theoretical categories, I Gelaki, Shlomo Sanmarco, Guillermo Quantum Algebra Representation Theory 18M20, 16T05, 17B37 Let $\mathscr {C}(G,H,ψ)$ be a finite group scheme-theoretical category over an algebraically closed field of characteristic $p\ge 0$ as introduced by the first author. For any indecomposable exact module category over $\mathscr {C}(G,H,ψ)$, we classify its simple objects and provide an expression for their projective covers, in terms of double cosets and projective representations of certain closed subgroup schemes, which upgrades a result by Ostrik for group-theoretical fusion categories. As a byproduct, we describe the simples and indecomposable projectives of $\mathscr {C}(G,H,ψ)$, and parametrize the Brauer-Piccard group of ${\rm Coh}(G)$ for any finite connected group scheme $G$. Finally, we apply our results to describe the blocks of the center of ${\rm Coh}(G)$. |
| title | On finite group scheme-theoretical categories, I |
| topic | Quantum Algebra Representation Theory 18M20, 16T05, 17B37 |
| url | https://arxiv.org/abs/2306.10205 |