Annals of Combinatorics

, Volume 16, Issue 1, pp 1–36 | Cite as

Words and Polynomial Invariants of Finite Groups in Non-Commutative Variables

  • Anouk Bergeron-Brlek
  • Christophe Hohlweg
  • Mike Zabrocki
Article
  • 70 Downloads

Abstract

Let V be a complex vector space with basis {x 1, x 2, . . . , x n } and G be a finite subgroup of GL(V). The tensor algebra T(V) over the complex is isomorphic to the polynomials in the non-commutative variables x 1, x 2, . . . , x n with complex coefficients. We want to give a combinatorial interpretation for the decomposition of T(V) into simple G-modules. In particular, we want to study the graded space of invariants in T(V) with respect to the action of G. We give a general method for decomposing the space T(V) into simple modules in terms of words in a Cayley graph of the group G. To apply the method to a particular group, we require a homomorphism from a subalgebra of the group algebra into the character algebra. In the case of G as the symmetric group, we give an example of this homomorphism from the descent algebra. When G is the dihedral group, we have a realization of the character algebra as a subalgebra of the group algebra. In those two cases, we have an interpretation for the graded dimensions and the number of free generators of the algebras of invariants in terms of those words.

Mathematics Subject Classification

05C25 05E10 20C30 20F55 20F10 

Keywords

non-commutative invariant theory representation theory finite groups symmetric group dihedral group Cayley graphs words 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bergeron N., Reutenauer C., Rosas M., Zabrocki M.: Invariants and coinvariants of the symmetric groups in noncommuting variables. Canad. J. Math. 60(2), 266–296 (2008)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Chauve, C., Goupil, A.: Combinatorial operators for Kronecker powers of representations S n. Sém. Lothar. Combin. 54, Art. B54j (2006)Google Scholar
  3. 3.
    Chevalley C.: Invariants of finite groups generated by reflections. Amer. J. Math. 77, 778–782 (1955)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Comtet L.: Advanced Combinatorics: The Art of Finite and Infinite Expansions. D. Reidel Publishing Co., Dordrecht (1974)MATHGoogle Scholar
  5. 5.
    Dicks, W., Formanek, E.: Poincaré series and a problem of S. Montgomery. Linear and Multilinear Algebra 12(1), 21–30 (1982/83)Google Scholar
  6. 6.
    Humphreys J.E.: Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics, 29. Cambridge University Press, Cambridge (1990)Google Scholar
  7. 7.
    Kharchenko V.K.: Algebras of invariants of free algebras. Algebra and Logic 17(4), 478–487 (1978)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Lane, D.R.: Free Algebras of Rank Two and Their Automorphisms. Ph.D thesis, Bedford College, London (1976)Google Scholar
  9. 9.
    MacMahon, P.A.: Combinatory Analysis. Vol. I, II. Cambridge University Press, Cambridge (1915/1916)Google Scholar
  10. 10.
    Molien, T.: Uber die Invarianten der linearen Substitutions gruppe. Sitz. Konig. Preuss. Akad. Wiss. 1152–1156 (1897)Google Scholar
  11. 11.
    Poirier S., Reutenauer C.: Algèbres de Hopf de tableaux. Ann. Sci. Math. Québec 19(1), 79–90 (1995)MathSciNetMATHGoogle Scholar
  12. 12.
    de Robinson G.B.: On the representations of the symmetric group. Amer. J. Math. 60(3), 745–760 (1938)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Rosas M.H., Sagan B.E.: Symmetric functions in noncommuting variables. Trans. Amer. Math. Soc. 358(1), 215–232 (2006)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Schensted C.: Longest increasing and decreasing subsequences. Canad. J.Math. 13, 179–191 (1961)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Shephard G.C., Todd J.A.: Finite unitary reflection groups. Canad. J. Math. 6, 274–304 (1954)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Solomon L.: A Mackey formula in the group ring of a Coxeter group. J. Algebra 41(2), 255–264 (1976)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Stanley R.P.: Invariants of finite groups and their applications to combinatorics. Bull. Amer. Math. Soc. (N.S.) 1(3), 475–511 (1979)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Wolf M.C.: Symmetric functions of non-commutative elements. Duke Math. J. 2(4), 626–637 (1936)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  • Anouk Bergeron-Brlek
    • 1
  • Christophe Hohlweg
    • 2
  • Mike Zabrocki
    • 1
  1. 1.Department of Mathematics and StatisticsYork UniversityTorontoCanada
  2. 2.LaCIM, Université du Québec à MontréalMontréal (Québec)Canada

Personalised recommendations