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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the irreducible representations of a finite semigroup


Authors: Olexandr Ganyushkin, Volodymyr Mazorchuk and Benjamin Steinberg
Journal: Proc. Amer. Math. Soc. 137 (2009), 3585-3592
MSC (2000): Primary 16G10, 20M30, 20M25
Published electronically: July 1, 2009
MathSciNet review: 2529864
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Abstract: Work of Clifford, Munn and Ponizovskiĭ parameterized the irreducible representations of a finite semigroup in terms of the irreducible representations of its maximal subgroups. Explicit constructions of the irreducible representations were later obtained independently by Rhodes and Zalcstein and by Lallement and Petrich. All of these approaches make use of Rees's theorem characterizing 0-simple semigroups up to isomorphism. Here we provide a short modern proof of the Clifford-Munn-Ponizovskiĭ result based on a lemma of J. A. Green, which allows us to circumvent the theory of 0-simple semigroups. A novelty of this approach is that it works over any base ring.


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Additional Information

Olexandr Ganyushkin
Affiliation: Department of Mechanics and Mathematics, Kyiv Taras Shevchenko University, 64, Volodymyrska Street, 01033, Kyiv, Ukraine
Email: ganiyshk@univ.kiev.ua

Volodymyr Mazorchuk
Affiliation: Department of Mathematics, Uppsala University, SE 471 06, Uppsala, Sweden – and – Department of Mathematics, University of Glasgow, University Gardens, Glasgow G12 8QW, United Kingdom
Email: mazor@math.uu.se, v.mazorchuk@maths.gla.ac.uk

Benjamin Steinberg
Affiliation: School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario K1S 5B6, Canada
Email: bsteinbg@math.carleton.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-09-09857-8
PII: S 0002-9939(09)09857-8
Keywords: Representations, simple modules, semigroups
Received by editor(s): December 17, 2007
Received by editor(s) in revised form: November 28, 2008
Published electronically: July 1, 2009
Additional Notes: The first author was supported in part by STINT
The second author was supported in part by the Swedish Research Council
The third author was supported in part by NSERC
Communicated by: Birge Huisgen-Zimmermann
Article copyright: © Copyright 2009 American Mathematical Society