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

Eaton, C. W., Eisele, F. and Livesey, M. (2020) Donovan’s conjecture, blocks with abelian defect groups and discrete valuation rings. Mathematische Zeitschrift, 295, pp. 249-264. (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 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.

• Item Type: Articles
• Additional Information: This research was supported by the EPSRC (Grant Nos. EP/M015548/1 and EP/M02525X/1).
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Eisele, Dr Florian
• Authors: Eaton, C. W., Eisele, F., and Livesey, M.
• College/School: College of Science and Engineering > School of Mathematics and Statistics
• Journal Name: Mathematische Zeitschrift
• Publisher: Springer
• ISSN: 0025-5874
• ISSN (Online): 1432-1823
• Published Online: 08 July 2019
• Copyright Holders: Copyright © 2019 The Authors
• First Published: First published in Mathematische Zeitschrift 295:249–264
• Publisher Policy: Reproduced under a Creative Commons License