Advertisement

Introduction

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

Abstract

A brief history of the Hopf algebra of quasisymmetric functions is given, along with their appearance in discrete geometry, representation theory and algebra. A discussion on how quasisymmetric functions simplify other algebraic functions is undertaken, and their appearance in areas such as probability, topology, and graph theory is also covered. Research on the dual algebra of noncommutative symmetric functions is touched on, as is a variety of extensions to quasisymmetric functions. What is known about the basis of quasisymmetric Schur functions is also addressed.

Key words

history of quasisymmetric functions history of quasisymmetric Schur functions 

References

  1. 2.
    Aguiar, M., Hsiao, S.: Canonical characters on quasi-symmetric functions and bivariate Catalan numbers. Electron. J. Combin. 11, Paper 15, 34 pp (2004/06)Google Scholar
  2. 3.
    Aguiar, M., Sottile, F.: Structure of the Malvenuto-Reutenauer Hopf algebra of permutations. Adv. Math. 191, 225–275 (2005)Google Scholar
  3. 4.
    Aguiar, M., Bergeron, N., Sottile, F.: Combinatorial Hopf algebras and generalized Dehn-Sommerville relations. Compos. Math. 142, 1–30 (2006)Google Scholar
  4. 5.
    Assaf, S.: On dual equivalence and Schur positivity. Preprint. arXiv:1107.0090v1[math.CO]Google Scholar
  5. 6.
    Aval, J.-C., Bergeron, F., Bergeron, N.: Ideals of quasi-symmetric functions and super-covariant polynomials for S n. Adv. Math. 181, 353–367 (2004)Google Scholar
  6. 7.
    Baker, A., Richter, B.: Quasisymmetric functions from a topological point of view. Math. Scand. 103, 208–242 (2008)Google Scholar
  7. 8.
    Baumann, P., Hohlweg, C.: A Solomon descent theory for the wreath products \(G \wr \mathfrak{S}_{n}\). Trans. Amer. Math. Soc. 360, 1475–1538 (2008)Google Scholar
  8. 9.
    Bender, E., Knuth, D.: Enumeration of plane partitions. J. Combin. Theory Ser. A 13, 40–54 (1972)Google Scholar
  9. 10.
    Bergeron, N., Zabrocki, M.: The Hopf algebras of symmetric functions and quasi-symmetric functions in non-commutative variables are free and co-free. J. Algebra Appl. 8, 581–600 (2009)Google Scholar
  10. 12.
    Bergeron, N., Mykytiuk, S., Sottile, F., van Willigenburg, S.: Noncommutative Pieri operators on posets. J. Combin. Theory Ser. A 91, 84–110 (2000)Google Scholar
  11. 13.
    Bergeron, N., Mykytiuk, S., Sottile, F., van Willigenburg, S.: Shifted quasi-symmetric functions and the Hopf algebra of peak functions. Discrete Math. 246, 57–66 (2002)Google Scholar
  12. 14.
    Bessenrodt, C., van Willigenburg, S.: Multiplicity free Schur, skew Schur, and quasisymmetric Schur functions. Preprint, to appear Ann. Comb. arXiv:1105.4212v2[math.CO]Google Scholar
  13. 15.
    Bessenrodt, C., Luoto, K., van Willigenburg, S.: Skew quasisymmetric Schur functions and noncommutative Schur functions. Adv. Math. 226, 4492–4532 (2011)Google Scholar
  14. 16.
    Billera, L., Brenti, F.: Quasisymmetric functions and Kazhdan-Lusztig polynomials. Israel J. Math. 184, 317–348 (2011)Google Scholar
  15. 17.
    Billera, L., Hsiao, S., van Willigenburg, S.: Peak quasisymmetric functions and Eulerian enumeration. Adv. Math. 176, 248–276 (2003)Google Scholar
  16. 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
  17. 19.
    Billera, L., Jia, N., Reiner, V.: A quasisymmetric function for matroids. European J. Combin. 30, 1727–1757 (2009)Google Scholar
  18. 20.
    Billey, S., Haiman, M.: Schubert polynomials for the classical groups. J. Amer. Math. Soc. 8, 443–482 (1995)Google Scholar
  19. 21.
    Blessenohl, D., Schocker, M.: Noncommutative character theory of the symmetric group. Imperial College Press, London (2005)Google Scholar
  20. 22.
    Buchstaber,V., Erokhovets, N.: Polytopes, Hopf algebras and Quasi-symmetric functions. Preprint. arXiv:1011.1536v1[math.CO]Google Scholar
  21. 23.
    Chow, T.: Descents, quasi-symmetric functions, Robinson-Schensted for posets, and the chromatic symmetric function. J. Algebraic Combin. 10, 227–240 (1999)Google Scholar
  22. 24.
    Derksen, H.: Symmetric and quasi-symmetric functions associated to polymatroids. J. Algebraic Combin. 30, 43–86 (2009)Google Scholar
  23. 25.
    Dimakis, A., Müller-Hoissen, F.: Quasi-symmetric functions and the KP hierarchy. J. Pure Appl. Algebra 214, 449–460 (2010)Google Scholar
  24. 26.
    Duchamp, G., Hivert, F., Thibon, J.-Y.: Noncommutative symmetric functions. VI. Free quasi-symmetric functions and related algebras. Internat. J. Algebra Comput. 12, 671–717 (2002)Google Scholar
  25. 27.
    Ehrenborg, R.: On posets and Hopf algebras. Adv. Math. 119, 1–25 (1996)Google Scholar
  26. 28.
    Ehrenborg, R., Readdy, M.: The Tchebyshev transforms of the first and second kind. Ann. Comb. 14, 211–244 (2010)Google Scholar
  27. 29.
    Ferreira, J.: A Littlewood-Richardson Type Rule for Row-Strict Quasisymmetric Schur Functions. Preprint. arXiv:1102.1458v1[math.CO]Google Scholar
  28. 30.
    Foissy, L.: Faà di Bruno subalgebras of the Hopf algebra of planar trees from combinatorial Dyson-Schwinger equations. Adv. Math. 218, 136–162 (2008)Google Scholar
  29. 31.
    Fulton, W.: Young tableaux. With applications to representation theory and geometry. Cambridge University Press, Cambridge (1997)Google Scholar
  30. 32.
    Garsia, A., Reutenauer, C.: A decomposition of Solomon’s descent algebra. Adv. Math. 77, 189–262 (1989)Google Scholar
  31. 33.
    Garsia, A., Wallach, N.: \(r -\text{QSym}\) is free over \(\text{Sym}\). J. Combin. Theory Ser. A 114, 704–732 (2007)Google Scholar
  32. 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
  33. 35.
    Gessel, I.: Multipartite P-partitions and inner products of skew Schur functions. Contemp. Math. 34, 289–301 (1984)Google Scholar
  34. 36.
    Gessel, I., Reutenauer, C.: Counting permutations with given cycle structure and descent set. J. Combin. Theory Ser. A 64, 189–215 (1993)Google Scholar
  35. 37.
    Gnedin, A., Olshanski, G.: Coherent permutations with descent statistic and the boundary problem for the graph of zigzag diagrams. Int. Math. Res. Not. Art. ID 51968, 39 pp (2006)Google Scholar
  36. 38.
    Haglund, J., Haiman, M., Loehr, N.: A combinatorial formula for Macdonald polynomials. J. Amer. Math. Soc. 18, 735–761 (2005)Google Scholar
  37. 39.
    Haglund, J., Luoto, K., Mason, S., van Willigenburg, S.: Refinements of the Littlewood-Richardson rule. Trans. Amer. Math. Soc. 363, 1665–1686 (2011)Google Scholar
  38. 40.
    Haglund, J., Luoto, K., Mason, S., van Willigenburg, S.: Quasisymmetric Schur functions. J. Combin. Theory Ser. A 118, 463–490 (2011)Google Scholar
  39. 41.
    Hazewinkel, M.: The Leibniz-Hopf algebra and Lyndon words. CWI Report Dept. of Analysis, Algebra and Geometry. AM-R9612 (1996)Google Scholar
  40. 42.
    Hazewinkel, M.: Symmetric functions, noncommutative symmetric functions, and quasisymmetric functions. Acta Appl. Math. 75, 55–83 (2003)Google Scholar
  41. 43.
    Hazewinkel, M.: Symmetric functions, noncommutative symmetric functions and quasisymmetric functions. II. Acta Appl. Math. 85, 319–340 (2005)Google Scholar
  42. 44.
    Hazewinkel, M.: Explicit polynomial generators for the ring of quasisymmetric functions over the integers. Acta Appl. Math. 109, 39–44 (2010)Google Scholar
  43. 45.
    Hersh, P., Hsiao, S.: Random walks on quasisymmetric functions. Adv. Math. 222, 782–808 (2009)Google Scholar
  44. 46.
    Hivert, F.: Hecke algebras, difference operators, and quasi-symmetric functions. Adv. Math. 155, 181–238 (2000)Google Scholar
  45. 47.
    Hoffman, M.: (Non)commutative Hopf algebras of trees and (quasi)symmetric functions. In: Renormalization and Galois theories, pp. 209–227. IRMA Lect. Math. Theor. Phys., 15, Eur. Math. Soc., Zürich (2009)Google Scholar
  46. 48.
    Hsiao, S., Karaali, G.: Multigraded combinatorial Hopf algebras and refinements of odd and even subalgebras. J. Algebraic Combin. 34, 451–506 (2011)Google Scholar
  47. 49.
    Hsiao, S., Petersen, (T.) K.: Colored posets and colored quasisymmetric functions. Ann. Comb. 14, 251–289 (2010)Google Scholar
  48. 50.
    Humpert, B.: A quasisymmetric function generalization of the chromatic symmetric function. Electron. J. Combin. 18, Paper 31, 13 pp (2011)Google Scholar
  49. 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
  50. 52.
    Kwon, J.-H.: Crystal graphs for general linear Lie superalgebras and quasi-symmetric functions. J. Combin. Theory Ser. A 116, 1199–1218 (2009)Google Scholar
  51. 53.
    Lam, T., Pylyavskyy, P.: P-partition products and fundamental quasi-symmetric function positivity. Adv. in Appl. Math. 40, 271–294 (2008)Google Scholar
  52. 54.
    Lascoux, A., Novelli, J.-C., Thibon, J.-Y.: Noncommutative symmetric functions with matrix parameters. Preprint. arXiv:1110.3209v1[math.CO]Google Scholar
  53. 55.
    Lauve, A., Mason, S.: \(\text{QSym}\) over \(\text{Sym}\) has a stable basis. J. Combin. Theory Ser. A 118, 1661–1673 (2011)Google Scholar
  54. 57.
    Loehr, N., Warrington, G.: Quasisymmetric expansions of Schur-function plethysms. Proc. Amer. Math. Soc. 140, 1159–1171 (2012)Google Scholar
  55. 58.
    Loehr, N., Serrano, L., Warrington, G.: Transition matrices for symmetric and quasisymmetric Hall-Littlewood polynomials. Preprint. arXiv:1202.3411v1[math.CO]Google Scholar
  56. 59.
    Luoto, K.: A matroid-friendly basis for the quasisymmetric functions. J. Combin. Theory Ser. A 115, 777–798 (2008)Google Scholar
  57. 60.
    Macdonald, I.: Symmetric functions and Hall polynomials. Second edition. Oxford University Press, New York (1995)Google Scholar
  58. 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
  59. 62.
    Malvenuto, C., Reutenauer, C.: Duality between quasi-symmetric functions and the Solomon descent algebra. J. Algebra 177, 967–982 (1995)Google Scholar
  60. 65.
    Mason, S., Remmel, J.: Row-strict quasisymmetric Schur functions. Preprint. arXiv:1110.4014v1[math.CO]Google Scholar
  61. 66.
    McNamara, P.: EL-labelings, supersolvability and 0-Hecke algebra actions on posets. J. Combin. Theory Ser. A 101, 69–89 (2003)Google Scholar
  62. 67.
    Menous, F., Novelli, J.-C., Thibon, J.-Y.: Mould calculus, polyhedral cones, and characters of combinatorial Hopf algebras. Preprint. arXiv:1109.1634v2[math.CO]Google Scholar
  63. 70.
    Novelli, J.-C., Schilling, A.: The forgotten monoid. In: Combinatorial representation theory and related topics, pp. 71–83. RIMS Kokyuroku Bessatsu B8, Kyoto (2008)Google Scholar
  64. 71.
    Reutenauer, C.: Free Lie algebras. Oxford University Press, New York (1993)Google Scholar
  65. 72.
    Sagan, B.: The symmetric group. Representations, combinatorial algorithms, and symmetric functions. Second edition. Springer-Verlag, New York (2001)Google Scholar
  66. 74.
    Shareshian, J., Wachs, M.: Eulerian quasisymmetric functions. Adv. Math. 225, 2921–2966 (2010)Google Scholar
  67. 75.
    Shareshian, J., Wachs, M.: Chromatic quasisymmetric functions and Hessenberg varieties. Preprint. arXiv:1106.4287v3[math.CO]Google Scholar
  68. 76.
    Solomon, L.: A Mackey formula in the group ring of a Coxeter group. J. Algebra 41, 255–264 (1976)Google Scholar
  69. 77.
    Stanley, R.: Ordered structures and partitions. Mem. Amer. Math. Soc. 119, (1972)Google Scholar
  70. 78.
    Stanley, R.: On the number of reduced decompositions of elements of Coxeter groups. European J. Combin. 5, 359–372 (1984)Google Scholar
  71. 79.
    Stanley, R.: A symmetric function generalization of the chromatic polynomial of a graph. Adv. Math. 111, 166–194 (1995)Google Scholar
  72. 80.
    Stanley, R.: Flag-symmetric and locally rank-symmetric partially ordered sets. Electron. J. Combin. 3, Paper 6, 22 pp (1996)Google Scholar
  73. 81.
    Stanley, R.: Enumerative combinatorics. Vol. 2. Cambridge University Press, Cambridge (1999)Google Scholar
  74. 82.
    Stanley, R.: Generalized riffle shuffles and quasisymmetric functions. Ann. Comb. 5, 479–491 (2001)Google Scholar
  75. 83.
    Stembridge, J.: Enriched P-partitions. Trans. Amer. Math. Soc. 349, 763–788 (1997)Google Scholar
  76. 85.
    Szczesny, M.: Colored trees and noncommutative symmetric functions. Electron. J. Combin. 17, Note 19, 10 pp (2010)Google Scholar
  77. 87.
    Zhao, W.: A family of invariants of rooted forests. J. Pure Appl. Algebra 186, 311–327 (2004)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

Personalised recommendations