TAILIEUCHUNG - Đề tài " Derived equivalences for symmetric groups and sl2-categorification "

We define and study sl2 -categorifications on abelian categories. We show in particular that there is a self-derived (even homotopy) equivalence categorifying the adjoint action of the simple reflection. We construct categorifications 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 Brou´’s abelian defect group conjecture for symmetric groups. e We give similar results for general linear groups over finite fields. . | Annals of Mathematics Derived equivalences for symmetric groups and sl2-categorification By Joseph Chuang and Rapha el Rouquier Annals of Mathematics 167 2008 245-298 Derived equivalences for symmetric groups and sl2-categorification By Joseph Chuang and Raphael Rouquier Abstract 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 Broue 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 glra C and for rational representations of general linear groups over Fp 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. Contents 1. Introduction 2. Notation 3. Affine Hecke algebras . Definitions . Totally ramified central character . Quotients 4. Reminders . Adjunctions . Representations of sl2 5. sl2-categorification . Weak categorifications . Categorifications . Minimal categorification . Link with affine Hecke algebras . Decomposition of E F The first author was supported in this research by the Nuffield Foundation NAL 00352 G and the EPSRC GR R91151 01 . 246 JOSEPH CHUANG AND RAPHAEL ROUQUIER 6. Categorification of the reflection . Rickard s complexes . Derived equivalence from the simple reflection . Equivalences for the minimal categorification 7. Examples . Symmetric groups . Cyclotomic Hecke algebras . General linear groups over a finite field . .

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