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Rationality of blocks of quasi-simple finite groups


Authors: Niamh Farrell and Radha Kessar
Journal: Represent. Theory 23 (2019), 325-349
MSC (2010): Primary 20C20; Secondary 20C33
DOI: https://doi.org/10.1090/ert/530
Published electronically: September 30, 2019
MathSciNet review: 4013115
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Abstract: Let $ \ell $ be a prime number. We show that the Morita Frobenius number of an $ \ell $-block of a quasi-simple finite group is at most $ 4$ and that the strong Frobenius number is at most $ 4 \vert D\vert^2!$, where $ D$ denotes a defect group of the block. We deduce that a basic algebra of any block of the group algebra of a quasi-simple finite group over an algebraically closed field of characteristic $ \ell $ is defined over a field with $ \ell ^a $ elements for some $ a \leq 4 $. We derive consequences for Donovan's conjecture. In particular, we show that Donovan's conjecture holds for $ \ell $-blocks of special linear groups.


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Additional Information

Niamh Farrell
Affiliation: Fachbereich Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany
Email: farrell@mathematik.uni-kl.de

Radha Kessar
Affiliation: Department of Mathematics, City, University of London, Northampton Square, EC1V 0HB London, United Kingdom
Email: radha.kessar.1@city.ac.uk

DOI: https://doi.org/10.1090/ert/530
Received by editor(s): July 4, 2018
Received by editor(s) in revised form: May 24, 2019
Published electronically: September 30, 2019
Additional Notes: This article was partly written while the authors were visiting the Mathematical Sciences Research Institute in Berkeley, California in Spring 2018 for the programme Group Representation Theory and Applications supported by the National Science Foundation under Grant No. DMS-1440140.
The first author also gratefully acknowledges financial support from the DFG project SFB-TRR 195
Article copyright: © Copyright 2019 American Mathematical Society