Derived equivalences for symmetric groups and sl\_2-categorification

TL;DR

This paper develops sl_2-categorifications on abelian categories, establishing derived equivalences for blocks of symmetric groups and general linear groups, and confirming conjectures related to Broué's abelian defect group conjecture.

Contribution

It introduces new sl_2-categorifications and proves derived equivalences between blocks of symmetric and general linear groups, extending to cyclotomic Hecke algebras and category O.

Findings

Blocks with isomorphic defect groups are splendidly Rickard equivalent.

Derived equivalences confirm Broué's conjecture for symmetric groups.

Categorifications for category O and rational representations lead to new equivalences.

Abstract

We define and study 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 Brou\'e'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 gl\_n(C) and for rational representations of general linear groups over an algebraically closed field of characteristic 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.