Donovan\u2019s conjecture, blocks with abelian defect groups and discrete valuation rings

Charles W Eaton,Florian Eisele,Michael Livesey

doi:10.1007/s00209-019-02354-1

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 {mathcal {O}}. Consequences are that Donovan’s conjecture holds for {mathcal {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 {mathcal {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 {mathcal {O}}-blocks is a consequence of conjectures predicting bounds on the {mathcal {O}}-Frobenius number and on the Cartan invariants, as was proved by Kessar for blocks defined over an algebraically closed field.

Highlights

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 reduces to bounding Cartan invariants of blocks of quasisimple groups
We have shown that for abelian p-groups Conjecture 1.1 is equivalent to 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

Summary

Introduction

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 reduces to bounding Cartan invariants of blocks of quasisimple groups Such bounds are known to hold for 2-blocks with abelian defect groups. The results of this paper hold over either choice (see Remark 4.7 for the latter case), but in light of the results of [9] the former seems the best setting for Donovan’s conjecture

Strong O-Frobenius and O-Morita-Frobenius numbers

Morita-Frobenius numbers and Donovan’s conjecture

Reductions for Donovan’s conjecture

Bounding Cartan invariants