Abstract
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.
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References
Aguiar, M.: Infinitesimal Hopf algebras and the cd-index of polytopes. Discrete Comput. Geom. 27, 3–28 (2002)
Aguiar, M., Bergeron, N., Sottile, F.: Combinatorial Hopf algebras and generalized Dehn-Sommerville relations. Compos. Math. 142, 1–30 (2006)
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)
Billera, L., Thomas, H., van Willigenburg, S.: Decomposable compositions, symmetric quasisymmetric functions and equality of ribbon Schur functions. Adv. Math. 204, 204–240 (2006)
Ehrenborg, R.: On posets and Hopf algebras. Adv. Math. 119, 1–25 (1996)
Fulton, W.: Young tableaux. With applications to representation theory and geometry. Cambridge University Press, Cambridge (1997)
Gelfand, I., Krob, D., Lascoux, A., Leclerc, B., Retakh, V., Thibon, J.-Y.: Noncommutative symmetric functions. Adv. Math. 112, 218–348 (1995)
Gessel, I.: Multipartite P-partitions and inner products of skew Schur functions. Contemp. Math. 34, 289–301 (1984)
Hivert, F.: Hecke algebras, difference operators, and quasi-symmetric functions. Adv. Math. 155, 181–238 (2000)
Krob, D., Leclerc, B., Thibon, J.-Y.: Noncommutative symmetric functions. II. Transformations of alphabets. Internat. J. Algebra Comput. 7, 181–264 (1997)
Littlewood, D., Richardson, A.: Group characters and algebra. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 233, 99–141 (1934)
Macdonald, I.: Symmetric functions and Hall polynomials. Second edition. Oxford University Press, New York (1995)
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)
Malvenuto, C., Reutenauer, C.: Duality between quasi-symmetric functions and the Solomon descent algebra. J. Algebra 177, 967–982 (1995)
Malvenuto, C., Reutenauer, C.: Plethysm and conjugation of quasi-symmetric functions. Discrete Math. 193, 225–233 (1998)
Milnor, J., Moore, J.: On the structure of Hopf algebras. Ann. of Math. 81, 211–264 (1965)
Montgomery, S.: Hopf algebras and their actions on rings. American Mathematical Society, Providence (1993)
Sagan, B.: The symmetric group. Representations, combinatorial algorithms, and symmetric functions. Second edition. Springer-Verlag, New York (2001)
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)
Stanley, R.: Ordered structures and partitions. Mem. Amer. Math. Soc. 119, (1972)
Stanley, R.: Enumerative combinatorics. Vol. 2. Cambridge University Press, Cambridge (1999)
Stembridge, J.: Enriched P-partitions. Trans. Amer. Math. Soc. 349, 763–788 (1997)
Sweedler, M.: Hopf algebras. Benjamin, New York (1969)
Thomas, G.: On Schensted’s construction and the multiplication of Schur functions. Adv. Math. 30, 8–32 (1978)
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© 2013 Kurt Luoto, Stefan Mykytiuk, Stephanie van Willigenburg
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Luoto, K., Mykytiuk, S., van Willigenburg, S. (2013). Hopf algebras. In: An Introduction to Quasisymmetric Schur Functions. SpringerBriefs in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7300-8_3
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DOI: https://doi.org/10.1007/978-1-4614-7300-8_3
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