Hopf algebras

  • Kurt Luoto
  • Stefan Mykytiuk
  • Stephanie van Willigenburg
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)


We give the basic theory of graded Hopf algebras, and then illustrate the theory in detail with three examples: the Hopf algebra of symmetric functions, Sym, the Hopf algebra of quasisymmetric functions, QSym, and the Hopf algebra of noncommutative symmetric functions, NSym. In each case we describe pertinent bases, the product, the coproduct and the antipode. Once defined we see how Sym is a subalgebra of QSym, and a quotient of NSym. We also discuss the duality of QSym and NSym and a variety of automorphisms on each. We end by defining combinatorial Hopf algebras and discussing the role QSym plays as the terminal object in the category of all combinatorial Hopf algebras.

Key words

combinatorial Hopf algebras coproducts duality Hopf algebras noncommutative symmetric functions P-partitions products quasisymmetric functions symmetric functions 


  1. 1.
    Aguiar, M.: Infinitesimal Hopf algebras and the cd-index of polytopes. Discrete Comput. Geom. 27, 3–28 (2002)Google Scholar
  2. 4.
    Aguiar, M., Bergeron, N., Sottile, F.: Combinatorial Hopf algebras and generalized Dehn-Sommerville relations. Compos. Math. 142, 1–30 (2006)Google Scholar
  3. 11.
    Bergeron, N., Reutenauer, C., Rosas, M., Zabrocki, M.: Invariants and coinvariants of the symmetric groups in noncommuting variables. Canad. J. Math. 60, 266–296 (2008)Google Scholar
  4. 18.
    Billera, L., Thomas, H., van Willigenburg, S.: Decomposable compositions, symmetric quasisymmetric functions and equality of ribbon Schur functions. Adv. Math. 204, 204–240 (2006)Google Scholar
  5. 27.
    Ehrenborg, R.: On posets and Hopf algebras. Adv. Math. 119, 1–25 (1996)Google Scholar
  6. 31.
    Fulton, W.: Young tableaux. With applications to representation theory and geometry. Cambridge University Press, Cambridge (1997)Google Scholar
  7. 34.
    Gelfand, I., Krob, D., Lascoux, A., Leclerc, B., Retakh, V., Thibon, J.-Y.: Noncommutative symmetric functions. Adv. Math. 112, 218–348 (1995)Google Scholar
  8. 35.
    Gessel, I.: Multipartite P-partitions and inner products of skew Schur functions. Contemp. Math. 34, 289–301 (1984)Google Scholar
  9. 46.
    Hivert, F.: Hecke algebras, difference operators, and quasi-symmetric functions. Adv. Math. 155, 181–238 (2000)Google Scholar
  10. 51.
    Krob, D., Leclerc, B., Thibon, J.-Y.: Noncommutative symmetric functions. II. Transformations of alphabets. Internat. J. Algebra Comput. 7, 181–264 (1997)Google Scholar
  11. 56.
    Littlewood, D., Richardson, A.: Group characters and algebra. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 233, 99–141 (1934)Google Scholar
  12. 60.
    Macdonald, I.: Symmetric functions and Hall polynomials. Second edition. Oxford University Press, New York (1995)Google Scholar
  13. 61.
    MacMahon, P.: Combinatory analysis. Vol. I, II (bound in one volume). Reprint of An introduction to combinatory analysis (1920) and Combinatory analysis. Vol. I, II (1915, 1916). Dover Publications, Mineola (2004)Google Scholar
  14. 62.
    Malvenuto, C., Reutenauer, C.: Duality between quasi-symmetric functions and the Solomon descent algebra. J. Algebra 177, 967–982 (1995)Google Scholar
  15. 63.
    Malvenuto, C., Reutenauer, C.: Plethysm and conjugation of quasi-symmetric functions. Discrete Math. 193, 225–233 (1998)Google Scholar
  16. 68.
    Milnor, J., Moore, J.: On the structure of Hopf algebras. Ann. of Math. 81, 211–264 (1965)Google Scholar
  17. 69.
    Montgomery, S.: Hopf algebras and their actions on rings. American Mathematical Society, Providence (1993)Google Scholar
  18. 72.
    Sagan, B.: The symmetric group. Representations, combinatorial algorithms, and symmetric functions. Second edition. Springer-Verlag, New York (2001)Google Scholar
  19. 73.
    Schützenberger, M.-P.: La correspondance de Robinson. In: Combinatoire et représentation du groupe symétrique (Actes Table Ronde CNRS, Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976) Lecture Notes in Math., Vol. 579, pp. 59–113. Springer, Berlin (1977)Google Scholar
  20. 77.
    Stanley, R.: Ordered structures and partitions. Mem. Amer. Math. Soc. 119, (1972)Google Scholar
  21. 81.
    Stanley, R.: Enumerative combinatorics. Vol. 2. Cambridge University Press, Cambridge (1999)Google Scholar
  22. 83.
    Stembridge, J.: Enriched P-partitions. Trans. Amer. Math. Soc. 349, 763–788 (1997)Google Scholar
  23. 84.
    Sweedler, M.: Hopf algebras. Benjamin, New York (1969)Google Scholar
  24. 86.
    Thomas, G.: On Schensted’s construction and the multiplication of Schur functions. Adv. Math. 30, 8–32 (1978)Google Scholar

Copyright information

© Kurt Luoto, Stefan Mykytiuk, Stephanie van Willigenburg 2013

Authors and Affiliations

  • Kurt Luoto
    • 1
  • Stefan Mykytiuk
    • 2
  • Stephanie van Willigenburg
    • 1
  1. 1.Department of MathematicsUniversity of British ColumbiaVancouverCanada
  2. 2.Department of Mathematics and StatisticsYork UniversityTorontoCanada

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