Journal of Algebraic Combinatorics

, Volume 42, Issue 3, pp 763–791

# Skew row-strict quasisymmetric Schur functions

• Sarah K. Mason
• Elizabeth Niese
Article

## 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.

## Keywords

Quasisymmetric functions Schur functions Composition tableaux Littlewood–Richardson rule

## Notes

### Acknowledgments

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

## References

1. 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 TanakaGoogle Scholar
2. 2.
Aguiar, M., Bergeron, N., Sottile, F.: Combinatorial Hopf algebras and generalized Dehn–Sommerville relations. Compos. Math. 142(1), 1–30 (2006). doi:
3. 3.
Bessenrodt, C., Luoto, K., van Willigenburg, S.: Skew quasisymmetric Schur functions and noncommutative Schur functions. Adv. Math. 226(5), 4492–4532 (2011)
4. 4.
Ehrenborg, R.: On posets and Hopf algebras. Adv. Math. 119(1), 1–25 (1996)
5. 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. 6.
Fulton, W.: Young Tableaux, London Mathematical Society Student Texts, vol. 35. Cambridge University Press, Cambridge (1997). With applications to representation theory and geometryGoogle Scholar
7. 7.
Gessel, I.: Multipartite p-partitions and inner products of skew Schur functions. Contemp. Math 34, 289–301 (1984)
8. 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:
9. 9.
Haglund, J., Haiman, M., Loehr, N.: A combinatorial formula for non-symmetric Macdonald polynomials. Amer. J. Math. 130(2), 359–383 (2008)
10. 10.
Haglund, J., Luoto, K., Mason, S., van Willigenburg, S.: Quasisymmetric Schur functions. J. Comb. Theory Ser. A 118(2), 463–490 (2011). doi:
11. 11.
Hazewinkel, M., Gubareni, N., Kirichenko, V.V.: Algebras, Rings and Modules, Mathematical Surveys and Monographs, vol. 168. American Mathematical Society, Providence (2010). doi:. Lie algebras and Hopf algebras
12. 12.
Hivert, F.: Hecke algebras, difference operators, and quasi-symmetric functions. Adv. Math. 155(2), 181–238 (2000). doi:
13. 13.
Lam, T., Lauve, A., Sottile, F.: Skew Littlewood–Richardson Rules from Hopf Algebras. Arxiv preprint arXiv:0908.3714 (2009)
14. 14.
Luoto, K., Mykytiuk, S., van Willigenburg, S.: An Introduction to Quasisymmetric Schur Functions. Springer Briefs in Mathematics. Springer, New York (2013). doi:. Hopf algebras, quasisymmetric functions, and Young composition tableaux
15. 15.
Malvenuto, C., Reutenauer, C.: Duality between quasi-symmetric functions and the Solomon descent algebra. J. Algebra 177(3), 967–982 (1995). doi:
16. 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. 17.
Mason, S., Remmel, J.: Row-strict quasisymmetric Schur functions. Ann. Comb. 18(1), 127–148 (2014)
18. 18.
Schur, I.: Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen. Ph.D. thesis, Berlin (1901)Google Scholar
19. 19.
Stanley, R.P.: On the number of reduced decompositions of elements of coxeter groups. Eur. J. Comb. 5(4), 359–372 (1984). doi:
20. 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 FominGoogle Scholar
21. 21.
Sweedler, M.E.: Hopf Algebras. Mathematics Lecture Note Series. W. A. Benjamin Inc, New York (1969)Google Scholar
22. 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. 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:. Conference on Combinatorics and Physics (Los Alamos, NM)