Derived equivalences for symmetric groups and sl(2)-categorification
We define and study sl2-categorifications on abelian categories. We show in particular that there is a self-derived (even homotopy) equivalence cate-gorifying the adjoint action of the simple reflection. We construct categorifica-tions for blocks of symmetric groups and deduce that two blocks are sp...
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| Format: | Journal article |
| Language: | English |
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2008
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| _version_ | 1797082793599041536 |
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| author | Chuang, J Rouquier, R |
| author_facet | Chuang, J Rouquier, R |
| author_sort | Chuang, J |
| collection | OXFORD |
| description | We define and study sl2-categorifications on abelian categories. We show in particular that there is a self-derived (even homotopy) equivalence cate-gorifying the adjoint action of the simple reflection. We construct categorifica-tions for blocks of symmetric groups and deduce that two blocks are splendidly Rickard equivalent whenever they have isomorphic defect groups and we show that this implies Broué's abelian defect group conjecture for symmetric groups. We give similar results for general linear groups over finite fields. The constructions extend to cyclotomic Hecke algebras. We also construct categorifications for category O of gln(C) and for rational representations of general linear groups over F̄p, where we deduce that two blocks corresponding to weights with the same stabilizer under the dot action of the affine Weyl group have equivalent derived (and homotopy) categories, as conjectured by Rickard. |
| first_indexed | 2024-03-07T01:32:54Z |
| format | Journal article |
| id | oxford-uuid:9430bcf7-f441-4fc2-aec2-3cbc4afd7f71 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T01:32:54Z |
| publishDate | 2008 |
| record_format | dspace |
| spelling | oxford-uuid:9430bcf7-f441-4fc2-aec2-3cbc4afd7f712022-03-26T23:37:37ZDerived equivalences for symmetric groups and sl(2)-categorificationJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:9430bcf7-f441-4fc2-aec2-3cbc4afd7f71EnglishSymplectic Elements at Oxford2008Chuang, JRouquier, RWe define and study sl2-categorifications on abelian categories. We show in particular that there is a self-derived (even homotopy) equivalence cate-gorifying the adjoint action of the simple reflection. We construct categorifica-tions for blocks of symmetric groups and deduce that two blocks are splendidly Rickard equivalent whenever they have isomorphic defect groups and we show that this implies Broué's abelian defect group conjecture for symmetric groups. We give similar results for general linear groups over finite fields. The constructions extend to cyclotomic Hecke algebras. We also construct categorifications for category O of gln(C) and for rational representations of general linear groups over F̄p, where we deduce that two blocks corresponding to weights with the same stabilizer under the dot action of the affine Weyl group have equivalent derived (and homotopy) categories, as conjectured by Rickard. |
| spellingShingle | Chuang, J Rouquier, R Derived equivalences for symmetric groups and sl(2)-categorification |
| title | Derived equivalences for symmetric groups and sl(2)-categorification |
| title_full | Derived equivalences for symmetric groups and sl(2)-categorification |
| title_fullStr | Derived equivalences for symmetric groups and sl(2)-categorification |
| title_full_unstemmed | Derived equivalences for symmetric groups and sl(2)-categorification |
| title_short | Derived equivalences for symmetric groups and sl(2)-categorification |
| title_sort | derived equivalences for symmetric groups and sl 2 categorification |
| work_keys_str_mv | AT chuangj derivedequivalencesforsymmetricgroupsandsl2categorification AT rouquierr derivedequivalencesforsymmetricgroupsandsl2categorification |