Skip to main content
SpringerLink
Log in

Skew row-strict quasisymmetric Schur functions

  • Published:
Journal of Algebraic Combinatorics Aims and scope Submit manuscript

Abstract

Mason and Remmel introduced a basis for quasisymmetric functions known as the row-strict quasisymmetric Schur functions. This basis is generated combinatorially by fillings of composition diagrams that are analogous to the row-strict tableaux that generate Schur functions. We introduce a modification known as Young row-strict quasisymmetric Schur functions, which are generated by row-strict Young composition fillings. After discussing basic combinatorial properties of these functions, we define a skew Young row-strict quasisymmetric Schur function using the Hopf algebra of quasisymmetric functions and then prove this is equivalent to a combinatorial description. We also provide a decomposition of the skew Young row-strict quasisymmetric Schur functions into a sum of Gessel’s fundamental quasisymmetric functions and prove a multiplication rule for the product of a Young row-strict quasisymmetric Schur function and a Schur function.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

Similar content being viewed by others

References

  1. Abe, E.: Hopf Algebras, Cambridge Tracts in Mathematics, vol. 74. Cambridge University Press, Cambridge-New York (1980). Translated from the Japanese by Hisae Kinoshita and Hiroko Tanaka

  2. Aguiar, M., Bergeron, N., Sottile, F.: Combinatorial Hopf algebras and generalized Dehn–Sommerville relations. Compos. Math. 142(1), 1–30 (2006). doi:10.1112/S0010437X0500165X

    Article  MATH  MathSciNet  Google Scholar 

  3. Bessenrodt, C., Luoto, K., van Willigenburg, S.: Skew quasisymmetric Schur functions and noncommutative Schur functions. Adv. Math. 226(5), 4492–4532 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  4. Ehrenborg, R.: On posets and Hopf algebras. Adv. Math. 119(1), 1–25 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  5. Ferreira, J.: A Littlewood–Richardson type rule for row-strict quasisymmetric Schur functions. Disc. Math. Theor. Comput. Sci. Proc. AO, 329–338 (2011)

    Google Scholar 

  6. Fulton, W.: Young Tableaux, London Mathematical Society Student Texts, vol. 35. Cambridge University Press, Cambridge (1997). With applications to representation theory and geometry

  7. Gessel, I.: Multipartite p-partitions and inner products of skew Schur functions. Contemp. Math 34, 289–301 (1984)

    Article  MathSciNet  Google Scholar 

  8. Gessel, I.M., Reutenauer, C.: Counting permutations with given cycle structure and descent set. J. Comb. Theory Ser. A 64(2), 189–215 (1993). doi:10.1016/0097-3165(93)90095-P

    Article  MATH  MathSciNet  Google Scholar 

  9. Haglund, J., Haiman, M., Loehr, N.: A combinatorial formula for non-symmetric Macdonald polynomials. Amer. J. Math. 130(2), 359–383 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  10. Haglund, J., Luoto, K., Mason, S., van Willigenburg, S.: Quasisymmetric Schur functions. J. Comb. Theory Ser. A 118(2), 463–490 (2011). doi:10.1016/j.jcta.2009.11.002

    Article  MATH  Google Scholar 

  11. Hazewinkel, M., Gubareni, N., Kirichenko, V.V.: Algebras, Rings and Modules, Mathematical Surveys and Monographs, vol. 168. American Mathematical Society, Providence (2010). doi:10.1090/surv/168. Lie algebras and Hopf algebras

  12. Hivert, F.: Hecke algebras, difference operators, and quasi-symmetric functions. Adv. Math. 155(2), 181–238 (2000). doi:10.1006/aima.1999.1901

    Article  MATH  MathSciNet  Google Scholar 

  13. Lam, T., Lauve, A., Sottile, F.: Skew Littlewood–Richardson Rules from Hopf Algebras. Arxiv preprint arXiv:0908.3714 (2009)

  14. Luoto, K., Mykytiuk, S., van Willigenburg, S.: An Introduction to Quasisymmetric Schur Functions. Springer Briefs in Mathematics. Springer, New York (2013). doi:10.1007/978-1-4614-7300-8. Hopf algebras, quasisymmetric functions, and Young composition tableaux

  15. Malvenuto, C., Reutenauer, C.: Duality between quasi-symmetric functions and the Solomon descent algebra. J. Algebra 177(3), 967–982 (1995). doi:10.1006/jabr.1995.1336

    Article  MATH  MathSciNet  Google Scholar 

  16. Mason, S.: A decomposition of Schur functions and an analogue of the Robinson–Schensted–Knuth algorithm. Séminaire Lotharingien de Combinatoire 57(B57e) (2008). http://www.citebase.org/abstract?id=oai:arXiv.org:math/0604430

  17. Mason, S., Remmel, J.: Row-strict quasisymmetric Schur functions. Ann. Comb. 18(1), 127–148 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  18. Schur, I.: Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen. Ph.D. thesis, Berlin (1901)

  19. Stanley, R.P.: On the number of reduced decompositions of elements of coxeter groups. Eur. J. Comb. 5(4), 359–372 (1984). doi:10.1016/S0195-6698(84)80039-6

    Article  MATH  Google Scholar 

  20. Stanley, R.P.: Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62. Cambridge University Press, Cambridge (1999). With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin

  21. Sweedler, M.E.: Hopf Algebras. Mathematics Lecture Note Series. W. A. Benjamin Inc, New York (1969)

    Google Scholar 

  22. Tewari, V., Willigenburg, S.: Quasisymmetric Schur functions and modules of the 0-hecke algebra. In: DMTCS Proceedings 0(01) (2014). http://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/article/view/dmAT0111

  23. Uglov, D.: Skew Schur functions and Yangian actions on irreducible integrable modules of \(\widehat{\mathfrak{g l}}_N\). Ann. Comb. 4(3–4), 383–400 (2000). doi:10.1007/PL00001287. Conference on Combinatorics and Physics (Los Alamos, NM)

Download references

Acknowledgments

Both authors were partially supported by a Wake Forest University Collaboration Pilot Grant.

Author information

Authors and Affiliations

  1. Department of Mathematics, Wake Forest University, Winston-Salem, NC, USA

    Sarah K. Mason

  2. Department of Mathematics, Marshall University, Huntington, WV, USA

    Elizabeth Niese

Authors
  1. Sarah K. Mason
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Elizabeth Niese
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Elizabeth Niese.

Appendix

Appendix

See Table 1 and Fig. 18.

Table 1 The four basic types of composition tableaux
Full size table
Fig. 18
figure 18

Triple configuration for composition tableaux

Full size image

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mason, S.K., Niese, E. Skew row-strict quasisymmetric Schur functions. J Algebr Comb 42, 763–791 (2015). https://doi.org/10.1007/s10801-015-0601-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10801-015-0601-6

Keywords

Search

Navigation