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

2018/09/21 by Eaton, Charles W., Eisele, Florian, Livesey, Michael
#20C #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1809.08152

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.

Related

• Donovan's conjecture and blocks with abelian defect groups
• Towards Donovan's conjecture for abelian defect groups
• Donovan's conjecture and extensions by the centralizer of a defect group
• Morita equivalence classes of blocks with extraspecial defect groups p₊¹⁺²
• Blocks with normal abelian defect and abelian p' inertial quotient