A Note on Jing and Li’s Type \(\varvec{B}\) Quasisymmetric Schur Functions

  • Ezgi Kantarcı OğuzEmail author


In 2015, Jing and Li defined type B quasisymmetric Schur functions and conjectured that these functions have a positive, integral and unitriangular expansion into peak functions. We prove this conjecture, and refine their combinatorial model to give explicit expansions in monomial, fundamental and peak bases. We also show that these functions are not quasisymmetric Schur, Young quasisymmetric Schur or dual immaculate positive, and do not have a positive multiplication rule.


Quasisymmetric functions Tableau combinatorics Quasisymmetric Schur P- and Q-functions Peak composition tableaux 

Mathematics Subject Classification

05E05 05E10 20C33 



  1. 1.
    Aguiar, M., Bergeron, N., Sottile, F.: Combinatorial Hopf algebras and generalized Dehn-Sommerville relations. Compos. Math. 142(1), 1–30 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Allan, E.E., Hallam, J., Mason. S.K.: Dual immaculate quasisymmetric functions expand positively into Young quasisymmetric Schur functions. J. Combin. Theory Ser. A 157, 70–108 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Berg, C., Bergeron, N., Saliola, F., Serrano, L., Zabrocki, M.: A lift of the Schur and Hall-Littlewood bases to non-commutative symmetric functions. Canad. J. Math. 66(3), 525–565 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Billera, L.J., Jia, N., Reiner, V.: A quasisymmetric function for matroids. European J. Combin. 30(8), 1727–1757 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Gessel, I.M.: Multipartite \(P\)-partitions and inner products of skew Schur functions. In: Greene, C.(ed.) Combinatorics and Algebra, Boulder, Colo., 1983, Contemp. Math., Vol. 34, pp. 289–317, Amer. Math. Soc., Providence, RI (1984)Google Scholar
  6. 6.
    Gessel, I.M., Reutenauer, C.: Counting permutations with given cycle structure and descent set. J. Combin. Theroy Ser. A 64(2), 189–215 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Haglund, J., Luoto, K., Mason, S., van Willigenburg, S.: Quasisymmetric Schur functions. J. Combin. Theory Ser. A 118(2), 463–490 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Jing, N., Li, Y.: A lift of Schur’s Q-functions to the peak algebra. J. Combin. Theory Ser. A 135, 268–290 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kwon, J.H.: Crystal graphs for general linear Lie superalgebras and quasi-symmetric functions. J. Combin. Theory Ser. A 116(7), 1199–1218 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Luoto, K., Mykytiuk, S., van Willigenburg, S.: An Introduction to Quasisymmetric Schur Functions: Hopf Algebras, Quasisymmetric Functions, and Young Composition Tableaux. SpringerBriefs in Mathematics. Springer, New York (2013)CrossRefzbMATHGoogle Scholar
  11. 11.
    Macdonald, I.G.: Symmetric Functions and Hall Polynomials. Second Edition. Oxford Classic Texts in the Physical Sciences. The Clarendon Press, Oxford University Press, New York (2015)Google Scholar
  12. 12.
    Oğuz, E.K.: A type B analogue to ribbon tableaux. arXiv:1701.07497 (2017)
  13. 13.
    Stembridge, J.R.: Enriched \(P\)-partitions. Trans. Amer. Math. Soc. 349(2), 763–788 (1997)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA

Personalised recommendations