Advertisement

Annals of Combinatorics

, Volume 22, Issue 1, pp 167–199 | Cite as

Quasisymmetric (k, l)-Hook Schur Functions

  • Sarah K. Mason
  • Elizabeth Niese
Article
  • 25 Downloads

Abstract

We introduce a quasisymmetric generalization of Berele and Regev’s hook Schur functions and prove that these new quasisymmetric hook Schur functions decompose the hook Schur functions in a natural way. We examine the combinatorics of the quasisymmetric hook Schur functions, providing a relationship to Gessel’s fundamental quasisymmetric functions and an analogue of the Robinson-Schensted-Knuth algorithm. We also prove that the multiplication of quasisymmetric hook Schur functions with hook Schur functions behaves the same as the multiplication of quasisymmetric Schur functions with Schur functions.

Mathematics Subject Classification

05E05 

Keywords

quasisymmetric functions Schur functions tableaux RSK algorithm 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Allen, E., Hallam, J., Mason, S.: Dual immaculate quasisymmetric functions expand positively into Young quasisymmetric Schur functions. arXiv:1606.03519 (2016)
  2. 2.
    Berele, A., Regev, A.: Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras. Adv. Math. 64(2), 118–175 (1987)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Berele, A., Remmel, J.B.: Hook flag characters and their combinatorics. J. Pure Appl. Algebra 35(3), 225–245 (1985)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Haglund, J., Haiman, M., Loehr, N., Remmel, J.B., Ulyanov, A.: A combinatorial formula for the character of the diagonal coinvariants. Duke Math. J. 126(2), 195–232 (2005)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Haglund, J., Luoto, K., Mason, S., vanWilligenburg, S.: Quasisymmetric Schur functions. J. Combin. Theory Ser. A 118(2), 463–490 (2011)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Kwon, J.-H.: Crystal graphs for general linear Lie superalgebras and quasi-symmetric functions. J. Combin. Theory Ser. A 116(7), 1199–1218 (2009)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Lauve, A., Mason, S.: QSym over Sym has a stable basis. J. Combin. Theory Ser. A 118(5), 1661–1673 (2011)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Luoto, K., Mykytiuk, S., van Willigenburg, S.: An Introduction to Quasisymmetric Schur Functions: Hopf Algebras, Quasisymmetric Functions, and Young Composition Tableaux. Springer, New York (2013)CrossRefMATHGoogle Scholar
  9. 9.
    Macdonald, I.G.: Schur functions: theme and variations. In: Zeng, J. (ed.) Séminaire Lotharingien de Combinatoire (Saint-Nabor, 1992), Publ. Inst. Rech. Math. Av, Vol. 498, pp. 5-39. Univ. Louis Pasteur, Strasbourg (1992)Google Scholar
  10. 10.
    Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (1995)MATHGoogle Scholar
  11. 11.
    Mason, S.: A decomposition of Schur functions and an analogue of the Robinson-Schensted-Knuth algorithm. Sém. Lothar. Combin. 57(2006/08), Art. B57e (2008)Google Scholar
  12. 12.
    Mason, S., Niese, E.: Quasisymmetric (\(k, l\))-hook Schur functions. In: 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), Discrete Math. Theor. Comput. Sci. Proc., AT. pp. 229-240. Assoc. Discrete Math. Theor. Comput. Sci., Nancy (2014)Google Scholar
  13. 13.
    Mason, S., Niese, E.: Skew row-strict quasisymmetric Schur functions. J. Algebraic Combin. 42(3), 763–791 (2015)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Mason, S., Remmel, J.B.: Row-strict quasisymmetric Schur functions. Ann. Combin. 18(1), 127–148 (2014)Google Scholar
  15. 15.
    Remmel, J.B.: The combinatorics of (\(k, l\))-hook Schur functions. In: Greene, C. (ed.) Combinatorics and Algebra, Contemp. Math., Vol. 34, pp. 253-287. Amer. Math. Soc., Providence, RI (1984)Google Scholar
  16. 16.
    Remmel, J.B.: Permutation statistics and (\(k, l\))-hook Schur functions. Discrete Math. 67(3), 271–298 (1987)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Schur, I.: Über eine Klasse von Matrizen, die sich einer gegebenen Matrix zuordnen lassen. PhD thesis Universität Berlin, Berlin (1901)Google Scholar
  18. 18.
    Tewari, V., van Willigenburg, S.: Modules of the 0-Hecke algebra and quasisymmetric Schur functions. Adv. Math. 285, 1025–1065 (2015)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Weyl, H.: The Classical Groups: Their Invariants and Representations. Princeton University Press, Princeton, NJ (1939)MATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsWake Forest UniversityWinston-SalemUSA
  2. 2.Department of MathematicsMarshall UniversityHuntingtonUSA

Personalised recommendations