On the irreducible representations of a finite semigroup
HTML articles powered by AMS MathViewer
- 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
- PDF | Request permission
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
- Jorge Almeida, Stuart Margolis, Benjamin Steinberg, and Mikhail Volkov, Representation theory of finite semigroups, semigroup radicals and formal language theory, Trans. Amer. Math. Soc. 361 (2009), no. 3, 1429–1461. MR 2457405, DOI 10.1090/S0002-9947-08-04712-0
- Kenneth S. Brown, Semigroups, rings, and Markov chains, J. Theoret. Probab. 13 (2000), no. 3, 871–938. MR 1785534, DOI 10.1023/A:1007822931408
- Kenneth S. Brown, Semigroup and ring theoretical methods in probability, Representations of finite dimensional algebras and related topics in Lie theory and geometry, Fields Inst. Commun., vol. 40, Amer. Math. Soc., Providence, RI, 2004, pp. 3–26. MR 2057147
- A. H. Clifford, Matrix representations of completely simple semigroups, Amer. J. Math. 64 (1942), 327–342. MR 6551, DOI 10.2307/2371687
- A. H. Clifford, Basic representations of completely simple semigroups, Amer. J. Math. 82 (1960), 430–434. MR 116062, DOI 10.2307/2372967
- A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. Vol. I, Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I., 1961. MR 0132791
- E. Cline, B. Parshall, and L. Scott, Finite-dimensional algebras and highest weight categories, J. Reine Angew. Math. 391 (1988), 85–99. MR 961165
- Olexandr Ganyushkin and Volodymyr Mazorchuk, Classical finite transformation semigroups, Algebra and Applications, vol. 9, Springer-Verlag London, Ltd., London, 2009. An introduction. MR 2460611, DOI 10.1007/978-1-84800-281-4
- J. A. Green, On the structure of semigroups, Ann. of Math. (2) 54 (1951), 163–172. MR 42380, DOI 10.2307/1969317
- James A. Green, Polynomial representations of $\textrm {GL}_{n}$, Lecture Notes in Mathematics, vol. 830, Springer-Verlag, Berlin-New York, 1980. MR 606556
- Michael A. Arbib (ed.), Algebraic theory of machines, languages, and semigroups, Academic Press, New York-London, 1968. With a major contribution by Kenneth Krohn and John L. Rhodes. MR 0232875
- Gérard Lallement and Mario Petrich, Irreducible matrix representations of finite semigroups, Trans. Amer. Math. Soc. 139 (1969), 393–412. MR 242973, DOI 10.1090/S0002-9947-1969-0242973-3
- S. W. Margolis and B. Steinberg. The quiver of an algebra associated to the Mantaci-Reutenauer descent algebra and the homology of regular semigroups. Algebr. Represent. Theory, to appear.
- D. B. McAlister, Characters of finite semigroups, J. Algebra 22 (1972), 183–200. MR 301125, DOI 10.1016/0021-8693(72)90111-1
- W. D. Munn, On semigroup algebras, Proc. Cambridge Philos. Soc. 51 (1955), 1–15. MR 66355, DOI 10.1017/s0305004100029868
- W. D. Munn, Matrix representations of semigroups, Proc. Cambridge Philos. Soc. 53 (1957), 5–12. MR 82050, DOI 10.1017/s0305004100031935
- I. S. Ponizovskiĭ, On matrix representations of associative systems, Mat. Sb. (N.S.) 38(80) (1956), 241–260 (Russian). MR 0081292
- Mohan S. Putcha, Complex representations of finite monoids, Proc. London Math. Soc. (3) 73 (1996), no. 3, 623–641. MR 1407463, DOI 10.1112/plms/s3-73.3.623
- Mohan S. Putcha, Complex representations of finite monoids. II. Highest weight categories and quivers, J. Algebra 205 (1998), no. 1, 53–76. MR 1631310, DOI 10.1006/jabr.1997.7395
- D. Rees, On semi-groups, Proc. Cambridge Philos. Soc. 36 (1940), 387–400. MR 2893
- John Rhodes and Benjamin Steinberg, The $q$-theory of finite semigroups, Springer Monographs in Mathematics, Springer, New York, 2009. MR 2472427, DOI 10.1007/b104443
- John Rhodes and Yechezkel Zalcstein, Elementary representation and character theory of finite semigroups and its application, Monoids and semigroups with applications (Berkeley, CA, 1989) World Sci. Publ., River Edge, NJ, 1991, pp. 334–367. MR 1142387
- Marcel-Paul Schützenberger, Sur la représentation monomiale des demi-groupes, C. R. Acad. Sci. Paris 246 (1958), 865–867 (French). MR 95886
- M. P. Schützenberger, Sur le produit de concaténation non ambigu, Semigroup Forum 13 (1976/77), no. 1, 47–75 (French). MR 444824, DOI 10.1007/BF02194921
- Benjamin Steinberg, Möbius functions and semigroup representation theory, J. Combin. Theory Ser. A 113 (2006), no. 5, 866–881. MR 2231092, DOI 10.1016/j.jcta.2005.08.004
- Benjamin Steinberg, Möbius functions and semigroup representation theory. II. Character formulas and multiplicities, Adv. Math. 217 (2008), no. 4, 1521–1557. MR 2382734, DOI 10.1016/j.aim.2007.12.001
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