Algebras and Representation Theory

, Volume 14, Issue 1, pp 131–159 | Cite as

The Quiver of an Algebra Associated to the Mantaci-Reutenauer Descent Algebra and the Homology of Regular Semigroups

Article

Abstract

We develop the homology theory of the algebra of a regular semigroup, which is a particularly nice case of a quasi-hereditary algebra in good characteristic. Directedness is characterized for these algebras, generalizing the case of semisimple algebras studied by Munn and Ponizovksy. We then apply homological methods to compute (modulo group theory) the quiver of a right regular band of groups, generalizing Saliola’s results for a right regular band. Right regular bands of groups come up in the representation theory of wreath products with symmetric groups in much the same way that right regular bands appear in the representation theory of finite Coxeter groups via the Solomon-Tits algebra of its Coxeter complex. In particular, we compute the quiver of Hsiao’s algebra, which is related to the Mantaci-Reutenauer descent algebra.

Keywords

Quivers Descent algebras Regular semigroups Representation theory 

Mathematics Subject Classifications (2000)

20M25 16G10 05E99 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aguiar, M., Mahajan, S.: Coxeter groups and Hopf algebras. In: Fields Institute Monographs, vol. 23. With a Foreword by Nantel Bergeron. American Mathematical Society, Providence (2006)Google Scholar
  2. 2.
    Almeida, J.: Finite semigroups and universal algebra. In: Series in Algebra, vol. 3. Translated from the 1992 Portuguese original and revised by the author. World Scientific, River Edge (1994)Google Scholar
  3. 3.
    Almeida, J., Margolis, S., Steinberg, B., Volkov, M.: Representation theory of finite semigroups, semigroup radicals and formal language theory. Trans. Am. Math. Soc. 361(3), 1429–1461 (2009)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Assem, I., Simson, D., Skowroński, A.: Elements of the representation theory of associative algebras, vol. 1. In: London Mathematical Society Student Texts, vol. 65. Techniques of Representation Theory. Cambridge University Press, Cambridge (2006)Google Scholar
  5. 5.
    Auslander, M., Platzeck, M.I., Todorov, G.: Homological theory of idempotent ideals. Trans. Am. Math. Soc. 332(2), 667–692 (1992)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Baumann, P., Hohlweg, C.: A Solomon descent theory for the wreath products \(G\wr \mathfrak S_n\). Trans. Am. Math. Soc. 360(3), 1475–1538 (2008) (electronic)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Benson, D.J.: Representations and cohomology, I, 2nd edn. In: Cambridge Studies in Advanced Mathematics, vol. 30. Basic Representation Theory of Finite Groups and Associative Algebras. Cambridge University Press, Cambridge (1998)Google Scholar
  8. 8.
    Bidigare, P., Hanlon, P., Rockmore, D.: A combinatorial description of the spectrum for the Tsetlin library and its generalization to hyperplane arrangements. Duke Math. J. 99(1), 135–174 (1999)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Brown, K.S.: Semigroups, rings, and Markov chains. J. Theor. Probab. 13(3), 871–938 (2000)MATHCrossRefGoogle Scholar
  10. 10.
    Brown, K.S.: Semigroup and ring theoretical methods in probability. In: Representations of Finite Dimensional Algebras and Related Topics in Lie Theory and Geometry. Fields Inst. Commun., vol. 40, pp. 3–26. American Mathematical Society, Providence (2004)Google Scholar
  11. 11.
    Cameron, P.J.: Permutation groups. In: London Mathematical Society Student Texts, vol. 45. Cambridge University Press, Cambridge (1999)Google Scholar
  12. 12.
    Clifford, A.H., Preston, G.B.: The Algebraic Theory of Semigroups, vol. I. Mathematical Surveys, No. 7. American Mathematical Society, Providence (1961)MATHGoogle Scholar
  13. 13.
    Cline, E., Parshall, B., Scott, L.: Finite-dimensional algebras and highest weight categories. J. Reine Angew. Math. 391, 85–99 (1988)MATHMathSciNetGoogle Scholar
  14. 14.
    Cline, E., Parshall, B., Scott, L.: Stratifying endomorphism algebras. Mem. Am. Math. Soc. 124(591), viii+119 (1996)MathSciNetGoogle Scholar
  15. 15.
    Curtis, C.W., Reiner, I.: Representation Theory of Finite Groups and Associative Algebras. Reprint of the 1962 original, A Wiley-Interscience Publication. Wiley Classics Library. Wiley, New York (1988)Google Scholar
  16. 16.
    Dixon, J.D., Mortimer, B.: Permutation groups. In: Graduate Texts in Mathematics, vol. 163. Springer, New York (1996)Google Scholar
  17. 17.
    Drozd, Y.A., Kirichenko, V.V.: Finite-dimensional Algebras. Translated from the 1980 Russian original and with an appendix by Vlastimil Dlab. Springer, Berlin (1994)Google Scholar
  18. 18.
    Du, J., Lin, Z.: Stratifying algebras with near-matrix algebras. J. Pure Appl. Algebra 188(1–3), 59–72 (2004)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Eilenberg, S.: Automata, Languages, and Machines, vol. B. With two chapters (“Depth decomposition theorem” and “Complexity of semigroups and morphisms”) by Bret Tilson. Pure and Applied Mathematics, vol. 59. Academic, New York (1976)Google Scholar
  20. 20.
    Ganyushkin, O., Mazorchuk, V., Steinberg, B.: On the irreducible representations of a finite semigroup. Proc. Am. Math. Soc. 137, 3585–3592 (2009)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Green, J.A.: On the structure of semigroups. Ann. Math. (2) 54, 163–172 (1951)CrossRefGoogle Scholar
  22. 22.
    Green, J.A.: Polynomial representations of GLn. In: Lecture Notes in Mathematics, vol. 830. Springer, Berlin (1980)Google Scholar
  23. 23.
    Hilton, P.J., Stammbach, U.: A course in homological algebra, 2nd edn. In: Graduate Texts in Mathematics, vol. 4. Springer, New York (1997)Google Scholar
  24. 24.
    Hsiao, S.K.: A semigroup approach to wreath-product extensions of Solomon’s descent algebras. Electron. J. Comb. 16(1), Research Paper 21, 9 (2009)Google Scholar
  25. 25.
    Jespers, E., Wang, Q., Hereditary semigroup algebras. J. Algebra 229(2), 532–546 (2000)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Krohn, K., Rhodes, J.: Algebraic theory of machines. I. Prime decomposition theorem for finite semigroups and machines. Trans. Am. Math. Soc. 116, 450–464 (1965)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Krohn, K., Rhodes, J., Tilson, B.: Algebraic theory of machines, languages, and semigroups. In: Arbib, M.A. (ed.) With a Major Contribution by Kenneth Krohn and John L. Rhodes, chapters 1, 5–9. Academic, New York (1968)Google Scholar
  28. 28.
    Kuzmanovich, J., Teply, M.L.: Homological dimensions of semigroup rings. Commun. Algebra 25(9), 2817–2837 (1997)MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Mantaci, R., Reutenauer, C.: A generalization of Solomon’s algebra for hyperoctahedral groups and other wreath products. Commun. Algebra 23(1), 27–56 (1995)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Nico, W.R.: Homological dimension in semigroup algebras. J. Algebra 18, 404–413 (1971)MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Nico, W.R.: An improved upper bound for global dimension of semigroup algebras. Proc. Am. Math. Soc. 35, 34–36 (1972)MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Novelli, J.-C., Thibon, J.-Y.: Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions. arXiv:0806.3682
  33. 33.
    Ponizovskiĭ, I.S.: Some examples of semigroup algebras of finite representation type. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 160(Anal. Teor. Chisel i Teor. Funktsii. 8), 229–238, 302 (1987)Google Scholar
  34. 34.
    Putcha, M.S.: Complex representations of finite monoids. Proc. Lond. Math. Soc. (3) 73(3), 623–641 (1996)MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Putcha, M.S.: Complex representations of finite monoids. II. Highest weight categories and quivers. J. Algebra 205(1), 53–76 (1998)MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Putcha, M.S.: Semigroups and weights for group representations. Proc. Am. Math. Soc. 128(10), 2835–2842 (2000)MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Renner, L.E.: Linear algebraic monoids. In: Encyclopaedia of Mathematical Sciences, vol. 134. Invariant Theory and Algebraic Transformation Groups, V. Springer, Berlin (2005)Google Scholar
  38. 38.
    Rhodes, J., Steinberg, B.: The q-theory of finite semigroups. In: Springer Monographs in Mathematics. Springer, New York (2009)Google Scholar
  39. 39.
    Rhodes, J., Zalcstein, Y.: Elementary representation and character theory of finite semigroups and its application. In: Monoids and Semigroups with Applications (Berkeley, CA, 1989), pp. 334–367. World Sci. Publ., River Edge, NJ (1991)Google Scholar
  40. 40.
    Saliola, F.V.: The quiver of the semigroup algebra of a left regular band. Int. J. Algebra Comput. 17(8), 1593–1610 (2007)MATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Schocker, M.: The module structure of the Solomon-Tits algebra of the symmetric group. J. Algebra 301(2), 554–586 (2006)MATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Stanley, R.P.: Enumerative combinatorics, vol. 1. In: Cambridge Studies in Advanced Mathematics, vol. 49. With a foreword by Gian-Carlo Rota, Corrected reprint of the 1986 original. Cambridge University Press, Cambridge (1997)Google Scholar
  43. 43.
    Steinberg, B.: Möbius functions and semigroup representation theory. J. Comb. Theory Ser. A 113(5), 866–881 (2006)MATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    Steinberg, B.: Möbius functions and semigroup representation theory. II. Character formulas and multiplicities. Adv. Math. 217(4), 1521–1557 (2008)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of MathematicsBar Ilan UniversityRamat GanIsrael
  2. 2.School of Mathematics and StatisticsCarleton UniversityOttawaCanada

Personalised recommendations