## On the irreducible representations of a finite semigroup

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- by Olexandr Ganyushkin, Volodymyr Mazorchuk and Benjamin Steinberg
- Proc. Amer. Math. Soc.
**137**(2009), 3585-3592 - DOI: https://doi.org/10.1090/S0002-9939-09-09857-8
- Published electronically: July 1, 2009
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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.## References

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## Bibliographic 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
- MR Author ID: 353912
- 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
- MR Author ID: 633258
- Email: bsteinbg@math.carleton.ca
- 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
- © Copyright 2009 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**137**(2009), 3585-3592 - MSC (2000): Primary 16G10, 20M30, 20M25
- DOI: https://doi.org/10.1090/S0002-9939-09-09857-8
- MathSciNet review: 2529864