The paper under review presents the most far reaching advance in Broué’s Abelian defect conjecture, using entirely new and most ingenious techniques. The result was announced by the authors at least six years ago. Still the result is the most striking one in the subject.
An \(\mathfrak{sl}_2\)-categorification is defined, roughly speaking, as a pair of adjoint and exact endo-functors \((E,F)\) of an Artinian Noetherian Abelian \(k\)-category \(A\), so that the action of \(E\) and \(F\) becomes the action of the standard elements \(e\) and \(f\) of the Lie-algebra \(\mathfrak{sl}_2\) on the Grothendieck group of \(A\), and that simple objects of \(A\) corresponds to weight vectors.
By highly complicate and very intrinsic considerations the authors construct explicit \(\mathfrak{sl}_2\)-categorifications on \(A\) being a category of modules over some Hecke algebra construction. This construction is the technical core of the paper. Using the categorification the authors construct a complex whose action corresponds to the standard reflection on \(\mathfrak{sl}_2\), changing the two basis vectors of the underlying vector space.
As a consequence, using previous work of the first author andR. Kessar [Bull. Lond. Math. Soc. 34, No. 2, 174-184 (2002;Zbl 1033.20009)] the authors show that two blocks of symmetric groups with isomorphic defect groups are splendidly Rickard equivalent, in particular derived equivalent. This implies as a very special case Broué’s Abelian defect conjecture.

Reviewer: Alexander Zimmermann (Amiens)