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

Abstract

We give a reduction to quasisimple groups for Donovan's conjecture for blocks with abelian defect groups defined with respect to a suitable discrete valuation ring O. Consequences are that Donovan's conjecture holds for O-blocks with abelian defect groups for the prime two, and that, using recent work of Farrell and Kessar, for arbitrary primes Donovan's conjecture for O-blocks with abelian defect groups reduces to bounding the Cartan invariants of blocks of quasisimple groups in terms of the defect.A result of independent interest is that in general (i.e. for arbitrary defect groups) Donovan's conjecture for O-blocks is a consequence of conjectures predicting bounds on the O-Frobenius number and on the Cartan invariants, as was proved by Kessar for blocks defined over an algebraically closed field. 1 studying groups generated by the defect groups, was only known over a field. The second is that the reduction in [12] of Donovan's conjecture into two distinct conjectures was also only known over a field. The first problem was overcome by the second author in [7], and we resolve the second here, allowing us to reduce Donovan's conjecture for O-blocks with abelian defect groups to bounding, for quasisimple groups, the Cartan invariants and strong Frobenius number as defined in [4]. The results of [9] show that the strong Frobenius numbers of quasisimple groups are bounded in terms of the defect group, so Donovan's conjecture for abelian defect groups in fact reduces to bounding Cartan invariants of blocks of quasisimple groups. Such bounds are known to hold for 2-blocks with abelian defect groups.Our main result is as follows:Theorem 1.2. Let d be a non-negative integer. If there are functions s, c : N → N such that for all O-blocks B of quasisimple groups with abelian defect groups of order p d ′ dividing p d , sf O (B) ≤ s(d ′ ) and all Cartan invariants are at most c(d ′ ), then Donovan's conjecture holds for O-blocks with abelian defect groups of order p d . A straightforward consequence is that: Corollary 1.3. Donovan's conjecture (over O) holds for blocks with abelian defect groups if and only if it holds for blocks of quasisimple groups with abelian defect groups. By work of Farrell and Kessar in [9], we get a much more powerful consequence: Corollary 1.4. Let d be a non-negative integer. If there is a function c : N → N such that for all O-blocks B of quasisimple groups with abelian defect groups of order p d ′ dividing p d the Cartan invariants are at most c(d ′ ), then Donovan's conjecture holds for O-blocks with abelian defect groups of order p d .Hence we have shown that for abelian p-groups Conjecture 1.1 is equivalent to (the restriction to quasisimple groups of) the following apparently much weaker conjecture, which arose from a question of Brauer:Conjecture 1.5 (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 ).Remark 1.6. It actuall...

Donovan's conjecture and extensions by the centralizer of a defect group

“…contradicting the minimality of G. A similar argument using Theorem 1.2 shows the bound on strong O-Frobenius numbers. We have shown that the Cartan invariants and the strong O-Frobenius numbers of all blocks with defect group isomorphic to P are bounded, and so the result follows by [14,Corollary 3.11]. ✷…”

“…In [21] Kessar showed that the k-Donovan conjecture is equivalent to showing Conjecture 1.1 and that the Morita Frobenius number (defined in §2) of a block is bounded in terms of the order of the defect groups. Variations on the Morita Frobenius number for blocks defined over O were given in [13], including the strong O-Frobenius number sf O (B), and in [14] the analogue of Kessar's result was shown for blocks defined over O. In [11] Düvel reduced Conjecture 1.1 to quasisimple groups (although the result in [11] is not quite strong enough for our purposes as stated).…”

“…In order to make canonical choices of k and O, we choose k to be the algebraic closure of the field F p of p-elements and O to be the ring of Witt vectors for k. There is a discussion of this in [14]. When we need to differentiate the versions of Donovan conjecture, the R-Donovan conjecture will relate to blocks defined with respect to the ring R, where R may be k or O.…”

On the geometry of lattices and finiteness of Picard groups

“…This is particularly important since Külshammer [13] showed that Picard groups play an important role in both Donovan's conjecture and the classification of blocks of a given defect group. This theme is explored in [6,7]. In the way of actual computations, [11] determines the Picard group of the principal block of A 6 for p D 3, [8] determines Picard groups of almost all blocks of abelian 2-defect of rank three, and this will certainly not mark the end of the story.…”

On the geometry of lattices and finiteness of Picard groups

“…This is particularly important since Külshammer [Kül95] showed that Picard groups play an important role in both Donovan's conjecture and the classification of blocks of a given defect group. This theme is explored in [Eat16,EEL19]. In the way of actual computations, [HN04] determines the Picard group of the principal block of A 6 for p = 3, [EL18] determines Picard groups of almost all blocks of abelian 2-defect of rank three, and this will certainly not mark the end of the story.…”