Algebras and Representation Theory

, Volume 20, Issue 2, pp 379–431 | Cite as

A Uniform Generalization of Some Combinatorial Hopf Algebras

  • Jia Huang


We generalize the Hopf algebras of free quasisymmetric functions, quasisymmetric functions, noncommutative symmetric functions, and symmetric functions to certain representations of the category of finite Coxeter systems and its dual category. We investigate their connections with the representation theory of 0-Hecke algebras of finite Coxeter systems. Restricted to type B and D we obtain dual graded modules and comodules over the corresponding Hopf algebras in type A.


Representation of categories Free quasisymmetric function Quasisymmetric function Noncomutative symmetric function Symmetric function Malvenuto–Reutenauer algebra Descent algebra 0-Hecke algebra Induction Restriction Coxeter group Type B type D 


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  1. 1.
    Aguiar, M., Mahajan, S.: Coxeter groups and Hopf algebras Fields Institute Monographs, vol. 23. AMS, Providence, RI (2006)Google Scholar
  2. 2.
    Assem, I., Simson, D., Skowroński, A.: Elements of the representation theory of associative algebras, vol. 1: Techniques of representation theory London Mathematical Society Student Texts, vol. 65. Cambridge University Press, Cambridge (2006)Google Scholar
  3. 3.
    Barcelo, H., Ihrig, E.: Lattices of parabolic subgroups in connection with hyperplane arrangements. J. Alg. Combin. 9, 5–24 (1999)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bergeron, N., Li, H.: Algebraic structures on Grothendieck groups of a tower of algebras. J. Algebra 321, 2068–2084 (2009)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Björner, A., Brenti, F.: Combinatorics of coxeter groups Graduate Texts in Mathematics, vol. 231. Springer, New York (2005)Google Scholar
  6. 6.
    Björner, A., Wachs, M.: Generalized quotients in Coxeter groups. Trans. Amer. Math. Soc. 308, 1–37 (1988)Google Scholar
  7. 7.
    Carter, R.W.: Representation theory of the 0-Hecke algebra. J. Algebra 104, 89–103 (1986)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chow, C.-O.: Noncommutative symmetric functions of type B, M.I.T. Ph.D. thesis (2001)Google Scholar
  9. 9.
    Denton, T.: A combinatorial formula for orthogonal idempotents in the 0-Hecke algebra of the symmetric group. Electron. J. Combin. 18(P28), 20 (2011)MathSciNetMATHGoogle Scholar
  10. 10.
    Denton, T., Hivert, F., Schilling, A., Thiéry, N.: On the representation theory of finite J-trivial monoids. Sém. Lothar. Combin. B64d, 44 (2011)MathSciNetMATHGoogle Scholar
  11. 11.
    Duchamp, G., Hivert, F., Thibon, J.-Y.: Noncommutative symmetric functions VI: free quasi-symmetric functions and related algebras, Internat. J. Alg. Comput. 12, 671–717 (2002)CrossRefMATHGoogle Scholar
  12. 12.
    Fayers, M.: 0-Hecke algebras of finite Coxeter groups. J. Pure Appl. Algebra 199, 27–41 (2005)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Geck, M., Pfeiffer, G.: Characters of finite Coxeter groups and Iwahori-Hecke algebras. Lond. Math. Soc. Monographs New Series, vol. 21. Oxford University Press, New York (2000)Google Scholar
  14. 14.
    Geissinger, L., Kinch, D.: Representations of the hyperoctahedral groups. J. Algebra 53, 1–20 (1978)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Grinberg, D., Reiner, V.: Hopf algebras in Combinatorics, arXiv:1409.8356v3
  16. 16.
    Huang, J.: 0-Hecke algebra actions on coinvariants and flags. J. Algebraic Combin. 40, 245–278 (2014)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Huang, J.: 0-Hecke algebra action on the Stanley-Reisner ring of the Boolean algebra. Ann. Comb. 19, 293–323 (2015)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Huang, J.: A tableau approach to the representation theory of 0-Hecke algebras, to appear in Annals of CombinatoricsGoogle Scholar
  19. 19.
    Humphreys, J.E.: Reflection groups and Coxeter groups Cambridge Advanced Studies in Mathematics, vol. 29. Cambridge University Press, Cambridge (1990)Google Scholar
  20. 20.
    Krob, D., Thibon, J.-Y.: Noncommutative symmetric functions IV: Quantum linear groups and Hecke algebras at q=0. J. Algebraic Combin. 6, 339–376 (1997)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Lusztig, G.: Hecke algebras with unequal parameters, arXiv:math/0208154v2
  22. 22.
    Malvenuto, C., Reutenauer, C.: Duality between quasi-symmetric functions and the solomon descent algebra. J. Algebra 177, 967–982 (1995)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Miller, A.: Reflection arrangements and ribbon representations. Eur. J. Combin. 39, 24–56 (2014)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Norton, P.N.: 0-Hecke algebras. J. Austral. Math. Soc. 27, 337–357 (1979)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Novelli, J.-C., Thibon, J.-Y.: Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions. Discret. Math. 310, 3584–3606 (2010)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Reiner, V.: Quotients of Coxeter complexes and P-partitions. Mem. AMS. no. 460(95), 1–134 (1992)MATHGoogle Scholar
  27. 27.
    Solomon, L.: A decomposition of the group algebra of a finite Coxeter group. J. Algebra 9, 220–239 (1968)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Stanley, R.: Enumerative combinatorics, vol. 2. Cambridge University Press (1999)Google Scholar
  29. 29.
    Stembridge, J.R.: A short derivation of the Möbius function for the Bruhat order. J. Algebr. Comb. 25, 141–148 (2007)CrossRefMATHGoogle Scholar

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© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of Nebraska at KearneyKearneyUSA

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