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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2503.02212 |
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| _version_ | 1866913717833170944 |
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| author | Munasinghe, Dinushi Webster, Ben |
| author_facet | Munasinghe, Dinushi Webster, Ben |
| contents | It's known that many different blocks of $\mathbb{F}_pS_n$ for different values of $n$ are equivalent as categories, though the corresponding block algebras are almost never isomorphic. Thus, it is a challenging problem to give one particularly nice representative of this Morita equivalence class of algebras. This has been accomplished for the case of RoCK blocks through work of Chuang--Kessar, Turner, and Evseev--Kleshchev. In this paper, we give a new perspective on this problem, applying not just to RoCK blocks of $S_n$, but also to all blocks of Ariki--Koike algebras. We do this by considering steadied quotients of KLRW algebras: these algebras are a natural generalization of cyclotomic quotients, already related to $S_n$ and Ariki--Koike algebras in work of Brundan--Kleshchev. These algebras are defined by ``tilting'' the cyclotomic relations so that we kill the two-sided ideal defined by certain configurations on the left and right sides of our diagrams. We show a Morita equivalence between these algebras and blocks of Ariki-Koike algebras generalizing the work discussed above. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_02212 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Steadied Khovanov-Lauda-Rouquier algebras and local models for blocks Munasinghe, Dinushi Webster, Ben Representation Theory Rings and Algebras It's known that many different blocks of $\mathbb{F}_pS_n$ for different values of $n$ are equivalent as categories, though the corresponding block algebras are almost never isomorphic. Thus, it is a challenging problem to give one particularly nice representative of this Morita equivalence class of algebras. This has been accomplished for the case of RoCK blocks through work of Chuang--Kessar, Turner, and Evseev--Kleshchev. In this paper, we give a new perspective on this problem, applying not just to RoCK blocks of $S_n$, but also to all blocks of Ariki--Koike algebras. We do this by considering steadied quotients of KLRW algebras: these algebras are a natural generalization of cyclotomic quotients, already related to $S_n$ and Ariki--Koike algebras in work of Brundan--Kleshchev. These algebras are defined by ``tilting'' the cyclotomic relations so that we kill the two-sided ideal defined by certain configurations on the left and right sides of our diagrams. We show a Morita equivalence between these algebras and blocks of Ariki-Koike algebras generalizing the work discussed above. |
| title | Steadied Khovanov-Lauda-Rouquier algebras and local models for blocks |
| topic | Representation Theory Rings and Algebras |
| url | https://arxiv.org/abs/2503.02212 |