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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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A proof of the first Kac–Weisfeiler conjecture in large characteristics
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by Benjamin Martin, David Stewart and Lewis Topley; with an appendix by Akaki Tikaradze PDF
Represent. Theory 23 (2019), 278-293 Request permission

Abstract:

In 1971, Kac and Weisfeiler made two influential conjectures describing the dimensions of simple modules of a restricted Lie algebra $\mathfrak {g}$. The first predicts the maximal dimension of simple $\mathfrak {g}$-modules and in this paper we apply the Lefschetz Principle and classical techniques from Lie theory to prove this conjecture for all restricted Lie subalgebras of $\mathfrak {gl}_n(k)$ whenever $k$ is an algebraically closed field of sufficiently large characteristic $p$ (depending on $n$). As a consequence we deduce that the conjecture holds for the Lie algebra of an affine algebraic group scheme over any commutative ring, after specialising to an algebraically closed field of almost any characteristic.

In the appendix to this paper, written by Akaki Tikaradze, an alternative, short proof of the first Kac–Weisfeiler conjecture is given for the Lie algebra of a group scheme over a finitely generated ring $R \subseteq \mathbb {C}$, after base change to a field of large positive characteristic.

References
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Additional Information
  • Benjamin Martin
  • Affiliation: Department of Mathematics, University of Aberdeen, King’s College, Fraser Noble Building, Aberdeen AB24 3UE, United Kingdom
  • MR Author ID: 659870
  • Email: B.Martin@abdn.ac.uk
  • David Stewart
  • Affiliation: School of Mathematics and Statistics, University of Newcastle, Herschel Building, Newcastle, NE1 7RU, United Kingdom
  • MR Author ID: 884527
  • Email: david.stewart@ncl.ac.uk
  • Lewis Topley
  • Affiliation: School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury, CT2 7FS, United Kingdom
  • MR Author ID: 1048635
  • ORCID: 0000-0002-4701-4384
  • Email: L.Topley@kent.ac.uk
  • Akaki Tikaradze
  • Affiliation: Department of Mathematics, Mail Stop 942, University of Toledo, 2801 W. Bancroft Street, Toledo, Ohio 43606-3390
  • MR Author ID: 676866
  • Email: Akaki.Tikaradze@utoledo.edu
  • Received by editor(s): November 16, 2018
  • Received by editor(s) in revised form: November 18, 2018, December 21, 2018, and July 30, 2019
  • Published electronically: September 16, 2019
  • Additional Notes: The second author is the corresponding author
    The third author gratefully acknowledges the support of EPSRC grant number EP/N034449/1
  • © Copyright 2019 American Mathematical Society
  • Journal: Represent. Theory 23 (2019), 278-293
  • MSC (2010): Primary 17B50; Secondary 17B10, 17B35, 03C60
  • DOI: https://doi.org/10.1090/ert/529
  • MathSciNet review: 4007168