Derived equivalences for symmetric groups and $\mathfrak{s}\mathfrak{l}_2$-categorification

Abstract

We define and study ${\mathfrak{sl}}_2$-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 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 $\mathcal{O}$ of $\mathfrak{gl}_n(\mathbf{C})$ and for rational representations of general linear groups over $\bar{\mathbf{F}}_p$, 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.