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

Charles W. Eaton*, Florian Eisele, Michael Livesey

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We give a reduction to quasisimple groups for Donovan's conjecture for blocks with abelian defect groups dened 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.
Original languageEnglish
Pages (from-to)249-264
Number of pages16
JournalMathematische Zeitschrift
Volume295
Early online date8 Jul 2019
DOIs
Publication statusPublished - 2019

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