Donovan’s conjecture and blocks with abelian defect groups

Abstract

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 ).

BLOCKS WITH NORMAL ABELIAN DEFECT AND ABELIAN p′ INERTIAL QUOTIENT

“…We show that these blocks are isomorphic to their second Frobenius twist. By [8], bounding Frobenius numbers is a key step towards Donovan's conjecture; see for instance [4], [5]. We obtain further a complete description of the basic algebra of such a block over a field by means of quiver with relations.…”

Rationality of blocks of quasi-simple finite groups

“…Motivation for the second part of Theorem 1.1 comes from a recent reduction of Donovan's conjecture for blocks with abelian defect, proved by Eaton and Livesey [16]. The second part of Theorem 1.1 in combination with the above reduction result shows that in order to prove Donovan's conjecture for blocks with abelian defect groups, it remains to show that the weak Donovan conjecture holds for blocks of quasi-simple finite groups with abelian defect groups.…”

Donovan’s conjecture, blocks with abelian defect groups and discrete valuation rings

“…This is already given in [5]: The remainder of the proof of Theorem 1.2 now consists of observing that bounding the strong O-Frobenius numbers for reduced pairs implies a bound on the number of Morita equivalence classes amongst reduced pairs. In [5], this part of the reduction could only be achieved over k since it relied on the results of [12]. The results of the previous section remedy this.…”

“…The general strategy for the reduction for Donovan's conjecture is the same as that in [5], where the reduction proceeds in two steps. First it is shown that it suffices to consider reduced pairs, and then it is shown that in order to demonstrate the conjecture for reduced pairs, we need only consider quasisimple groups.…”

“…We now give the first part of the reduction, which is analogous to [5,Proposition 3.4] and based on [2]. We give a proof here for completeness.…”