We give a reduction of Donovan's conjecture for abelian groups to a similar statement for quasisimple groups. Consequently we show that Donovan's conjecture holds for abelian 2-groups. IntroductionLet (K, O, k) be a p-modular system with k algebraically closed. We are interested in the following conjecture in the case of abelian p-groups:Conjecture 1.1 (Donovan). Let P be a finite p-group. Then amongst all finite groups G and blocks B of kG with defect groups isomorphic to P there are only finitely many Morita equivalence classes.One approach to the conjecture is reduction to quasisimple groups followed by the application of the classification of finite simple groups. For example, in [5] it was proved that Donovan's conjecture holds for elementary abelian 2-groups in this way, following a partial reduction in [3]. The reason that this result could not be extended to arbitrary abelian 2-groups is that it was not known how Morita equivalence classes of blocks relate to those for blocks of normal subgroups of index p in general, with only the special case of a split extension being known by [9]. Our approach uses [7], where it was shown that for each p-group P , Donovan's conjecture is equivalent to both of the following conjectures holding, the first originating from a question of Brauer:Conjecture 1.2 (Weak Donovan). Let P be a finite p-group. Then there is c(P ) ∈ N such that if G is a finite group and B is a block of kG with defect groups isomorphic to P , then the entries of the Cartan matrix of B are at most c(P ).

We define a new invariant for a p-block of a finite group, the strong Frobenius number, which we use to address the problem of reducing Donovan's conjecture to normal subgroups of index p. As an application we use the strong Frobenius number to complete the proof of Donovan's conjecture for 2-blocks with abelian defect groups of rank at most 4 and for 2-blocks with abelian defect groups of order at most 64.

Chapter 1. Introduction ix 1.1. Broué's conjecture for symmetric groups and beyond ix 1.2. RoCK blocks for double covers of symmetric groups xii 1.3. Quiver Hecke superalgebras xvii Chapter 2. Background material 2.1. Basic notation 2.2. Graded superalgebras and supermodules 2.3. Combinatorics 2.4. Lie theory Chapter 3. Quiver Hecke superalgebras 3.1. Quiver Hecke superalgebras and a dimension formula 3.2. Further properties of quiver Hecke superalgebras 3.3. Cuspidality Chapter 4. RoCK blocks of quiver Hecke superalgebras 4.1. RoCK blocks 4.2. Gelfand-Graev truncation of Rδ and Z ρ,1 4.3. Gelfand-Graev truncation of Rdδ 4.4. Intertwiners 4.5. Affine zigzag relations Chapter 5. RoCK blocks of double covers of symmetric groups 5.1. Twisted wreath superproducts 5.2. Blocks of double covers 5.3. Kang-Kashiwara-Tsuchioka isomorphism 5.4. Broué's conjecture for RoCK blocks of double covers Chapter 6. Appendix. Some calculations in B 2 6.1. Some small rank computations 6.2. A commutation lemma 6.

We calculate examples of Picard groups for 2-blocks with abelian defect groups with respect to a complete discrete valuation ring. These include all blocks with abelian 2-groups of 2-rank at most three with the exception of the principal block of J1. In particular this shows directly that all such Picard groups are finite and Picent, the group of Morita auto-equivalences fixing the centre, is trivial. These are amongst the first calculations of this kind. Further we prove some general results concerning Picard groups of blocks with normal defect groups as well as some other cases.

In this paper we classify all blocks with defect group C2n ×C2 ×C2 up to Morita equivalence. Together with a recent paper of Wu, Zhang and Zhou, this completes the classification of Morita equivalence classes of 2-blocks with abelian defect groups of rank at most 3. The classification holds for blocks over a suitable discrete valuation ring as well as for those over an algebraically closed field. The case considered in this paper is significant because it involves comparison of Morita equivalence classes between a group and a normal subgroup of index 2, so requires novel reduction techniques which we hope will be of wider interest. We note that this also completes the classification of blocks with abelian defect groups of order dividing 16 up to Morita equivalence. A consequence is that Broue's abelian defect group conjecture holds for all blocks mentioned above.Proof. If D is elementary abelian, then this is by [19], [7] and [8]. If D ∼ = C 4 × C 4 , then see [9] where it is shown that there are only two Morita equivalence classes. If D ∼ = C 4 × C 2 × C 2 , then this is Corollary 3.3. In all other cases Aut(D) is a 2-group and so all blocks with that defect group are nilpotent. Corollary 3.5. Let G be a finite group and B a block of OG with defect group D of 2-rank at most three. Let b be the unique block of N G (D) with b G = B. Then B and b are derived equivalent.Proof. This follows immediately from the above corollaries, [9, 1.1] and [31].