Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. J. Almeida, S. Margolis, B. Steinberg, and M. Volkov.
    Representation theory of finite semigroups, semigroup radicals and formal language theory.
    Trans. Amer. Math. Soc., 361(3):1429-1461, 2009. MR 2457405
  • 2. K. S. Brown.
    Semigroups, rings, and Markov chains.
    J. Theoret. Probab., 13(3):871-938, 2000. MR 1785534 (2001e:60141)
  • 3. K. S. Brown.
    Semigroup and ring theoretical methods in probability.
    In Representations of finite dimensional algebras and related topics in Lie theory and geometry, volume 40 of Fields Inst. Commun., pages 3-26. Amer. Math. Soc., Providence, RI, 2004. MR 2057147 (2005b:60118)
  • 4. A. H. Clifford.
    Matrix representations of completely simple semigroups.
    Amer. J. Math., 64:327-342, 1942. MR 0006551 (4:4a)
  • 5. A. H. Clifford.
    Basic representations of completely simple semigroups.
    Amer. J. Math., 82:430-434, 1960. MR 0116062 (22:6857)
  • 6. A. H. Clifford and G. B. Preston.
    The algebraic theory of semigroups. Vol. I.
    Mathematical Surveys, No. 7. American Mathematical Society, Providence, RI, 1961. MR 0132791 (24:A2627)
  • 7. E. Cline, B. Parshall, and L. Scott.
    Finite-dimensional algebras and highest weight categories.
    J. Reine Angew. Math., 391:85-99, 1988. MR 961165 (90d:18005)
  • 8. O. Ganyushkin and V. Mazorchuk.
    Classical finite transformation semigroups, an introduction.
    Number 9 in Algebra and Applications. Springer, 2009. MR 2460611
  • 9. J. A. Green.
    On the structure of semigroups.
    Ann. of Math. (2), 54:163-172, 1951. MR 0042380 (13:100d)
  • 10. J. A. Green.
    Polynomial representations of $ {\rm GL}\sb{n}$, volume 830 of Lecture Notes in Mathematics.
    Springer-Verlag, Berlin, 1980. MR 606556 (83j:20003)
  • 11. K. Krohn, J. Rhodes, and B. Tilson.
    Algebraic theory of machines, languages, and semigroups.
    Edited by Michael A. Arbib. With a major contribution by Kenneth Krohn and John L. Rhodes. Academic Press, New York, 1968, Chapters 1, 5-9. MR 0232875 (38:1198)
  • 12. G. Lallement and M. Petrich.
    Irreducible matrix representations of finite semigroups.
    Trans. Amer. Math. Soc., 139:393-412, 1969. MR 0242973 (39:4300)
  • 13. 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.
  • 14. D. B. McAlister.
    Characters of finite semigroups.
    J. Algebra, 22:183-200, 1972. MR 0301125 (46:283)
  • 15. W. D. Munn.
    On semigroup algebras.
    Proc. Cambridge Philos. Soc., 51:1-15, 1955. MR 0066355 (16:561c)
  • 16. W. D. Munn.
    Matrix representations of semigroups.
    Proc. Cambridge Philos. Soc., 53:5-12, 1957. MR 0082050 (18:489g)
  • 17. I. S. Ponizovskiĭ.
    On matrix representations of associative systems.
    Mat. Sb. N.S., 38(80):241-260, 1956. MR 0081292 (18:378d)
  • 18. M. S. Putcha.
    Complex representations of finite monoids.
    Proc. London Math. Soc. (3), 73(3):623-641, 1996. MR 1407463 (97e:20093)
  • 19. M. S. Putcha.
    Complex representations of finite monoids. II. Highest weight categories and quivers.
    J. Algebra, 205(1):53-76, 1998. MR 1631310 (99f:20106)
  • 20. D. Rees.
    On semi-groups.
    Proc. Cambridge Philos. Soc., 36:387-400, 1940. MR 0002893 (2:127g)
  • 21. J. Rhodes and B. Steinberg.
    The $ q$-theory of finite semigroups.
    Springer Monographs in Mathematics. Springer, New York, 2009. MR 2472427
  • 22. J. Rhodes and Y. Zalcstein.
    Elementary representation and character theory of finite semigroups and its application.
    In Monoids and semigroups with applications (Berkeley, CA, 1989), pages 334-367. World Sci. Publ., River Edge, NJ, 1991. MR 1142387 (92k:20129)
  • 23. M.-P. Schützenberger.
    Sur la représentation monomiale des demi-groupes.
    C. R. Acad. Sci. Paris, 246:865-867, 1958. MR 0095886 (20:2384)
  • 24. M. P. Schützenberger.
    Sur le produit de concaténation non ambigu.
    Semigroup Forum, 13(1):47-75, 1976/77. MR 0444824 (56:3171)
  • 25. B. Steinberg.
    Möbius functions and semigroup representation theory.
    J. Combin. Theory Ser. A, 113(5):866-881, 2006. MR 2231092 (2007c:20144)
  • 26. B. Steinberg.
    Möbius functions and semigroup representation theory. II. Character formulas and multiplicities.
    Adv. Math., 217(4):1521-1557, 2008. MR 2382734 (2009e:20131)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 16G10, 20M30, 20M25

Retrieve articles in all journals with MSC (2000): 16G10, 20M30, 20M25

Additional Information

Olexandr Ganyushkin
Affiliation: Department of Mechanics and Mathematics, Kyiv Taras Shevchenko University, 64, Volodymyrska Street, 01033, Kyiv, Ukraine

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

Benjamin Steinberg
Affiliation: School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario K1S 5B6, Canada

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

American Mathematical Society