Towards Donovan's conjecture for abelian defect groups

Abstract

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.

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.…”

“…Let B be a block of a finite group algebra over O with a normal defect group P and abelian inertial quotient L. Then B is isomorphic to its second Frobenius twist and if O = k, then B is a matrix algebra over a quantised version of the group algebra of the semidirect product P ⋊ L. Remark 1.4. It seems unclear whether the same bound holds for strong Frobenius numbers, introduced by Eaton and Livesey in [5]. One issue is that we do not have a sufficiently explicit description of the automorphism ϕ of P ⋊ L constructed in Lemma 2.3 below.…”

The Picard Group of an Order and Külshammer Reduction

“…However, these questions are outside of the scope of this short article. Instead, we focus our attention on an application in modular representation theory which follows from Theorem 1.1 in a fairly straightforward manner, a generalisation of a theorem of Külshammer (see [11] [11] were not known to hold over O prevented them from being combined with the results on "strong Frobenius numbers" in [5], which only work over O. This was one of the obstacles in reducing Donovan's conjecture for blocks of abelian defect (defined over O) to blocks of quasi-simple groups, which will be the subject of [4] (in preparation).…”

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

“…The strong O-Frobenius number was introduced in [4], but we recall the definition and some of its main properties here. We also define the O-Morita-Frobenius number.…”

“…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.…”