One of the most striking problems in the modular representation theory of finite groups is Broué’s abelian defect group conjecture, which states as follows.
Let (K, O, k) be a “big enough” p-modular system, G a finite group, b a block of OG with defect group D, H = NG(D), and let c be the Brauer correspondent of b in H. If D is abelian then the derived categories of the block algebras bOG and cOH are equivalent as triangulated categories.
In this survey talk we shall present some very recent results on this problem. We are going to discuss various more involved forms of the conjecture, the relationship with Dade’s conjectures, methods of constructing Rickard equivalences, and cases when the conjecture has been solved.
1. S. Koshitani and N. Kunugi, Broué’s conjecture holds for principal 3-blocks with elementary abelian defect groups of order 9, preprint 2000.
2. A. Marcus, Twisted group algebras, normal subgroups, and derived equivalences, preprint 2000, to appear in Algebras and Representation Theory.
3. T. Okuyama, Derived equivalence in SL(2, q), preprint 2000.
4. R. Rouquier, Block theory via stable and Rickard equivalence, preprint 2000.