Abstract
We review and further develop a general approach to Schur positivity of symmetric functions based on the machinery of noncommutative Schur functions. This approach unifies ideas of Assaf, Lam, and Greene and the second author.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Assaf, S.H.: Dual equivalence graphs and a combinatorial proof of LLT and Macdonald positivity. arXiv:1005.3759v5 (2013)
Assaf, S.H.: Dual equivalence and Schur positivity, preprint (2014) (The content of this paper has been included in [3])
Assaf, S.H.: Dual equivalence graphs I: a new paradigm for Schur positivity. Forum Math. Sigma 3, e12 (2015)
Blasiak, J., Liu, R.I.: Kronecker coefficients and noncommutative super Schur functions. arXiv:1510.00644
Blasiak, J.: Haglund’s conjecture on 3-column Macdonald polynomials. Math. Z. 283, 601–628 (2016)
Blasiak, J.: What makes a D\(_0\) graph Schur positive? J. Algebr. Comb. (2016). doi:10.1007/s10801-016-0685-7
Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24, 235–265 (1997)
Carré, C., Leclerc, B.: Splitting the square of a Schur function into its symmetric and antisymmetric parts. J. Algebr. Comb. 4, 201–231 (1995)
Edelman, P., Greene, C.: Balanced tableaux. Adv. Math. 63, 42–99 (1987)
Fishel, S.: Statistics for special \(q, t\)-Kostka polynomials. Proc. Am. Math. Soc. 123, 2961–2969 (1995)
Fomin, S., Greene, C.: Noncommutative Schur functions and their applications. Discrete Math. 193, 179–200 (1998)
Fomin, S., Stanley, R.P.: Schubert polynomials and the nil-Coxeter algebra. Adv. Math. 103, 196–207 (1994)
Garsia, A.M., Haiman, M.: A graded representation model for Macdonald’s polynomials. Proc. Natl. Acad. Sci. USA 90, 3607–3610 (1993)
Gessel, I.M.: Multipartite \(P\)-partitions and inner products of skew Schur functions. Contemp. Math. 34, 289–317 (1984)
Grojnowski, I., Haiman, M.: Affine Hecke algebras and positivity of LLT and Macdonald polynomials, preprint (2007)
Haglund, J., Haiman, M., Loehr, N.: A combinatorial formula for Macdonald polynomials. J. Am. Math. Soc. 18, 735–761 (2005)
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, 195–232 (2005)
Haiman, M.: Combinatorics, symmetric functions, and Hilbert schemes. In: Mazur, B. (ed.) Current Developments in Mathematics, 2002, pp. 39–111. Int. Press, Somerville (2003)
Kashiwara, M., Tanisaki, T.: Parabolic Kazhdan–Lusztig polynomials and Schubert varieties. J. Algebra 249, 306–325 (2002)
Kazhdan, D., Lusztig, G.: Representations of Coxeter groups and Hecke algebras. Invent. Math. 53, 165–184 (1979)
Kirillov, A. N.: Notes on Schubert, Grothendieck and key polynomials. SIGMA 12, 034 (2016). doi:10.3842/SIGMA.2016.034
Lam, T.: Ribbon Schur operators. Eur. J. Comb. 29, 343–359 (2008)
Lapointe, L., Morse, J.: Tableaux Statistics for Two Part Macdonald Polynomials, Algebraic Combinatorics and Quantum Groups. World Scientific, River Edge (2003)
Lascoux, A., Leclerc, B., Thibon, J.-Y.: Ribbon tableaux, Hall–Littlewood functions, quantum affine algebras, and unipotent varieties. J. Math. Phys. 38, 1041–1068 (1997)
Lascoux, A., Schützenberger, M.-P.: Le monoïde plaxique, noncommutative structures in algebra and geometric combinatorics (Naples 1978). Quad. “Ricerca Sci.” 109, 129–156 (1981)
Lascoux, A., Schützenberger, M.-P.: Structure de Hopf de l’anneau de cohomologie et de l’anneau de Grothendieck d’une variété de drapeaux. Acad. C. R. Sci. Paris Sér I Math. 295, 629–633 (1982)
Leclerc, B., Thibon, J.-Y.: Littlewood–Richardson coefficients and Kazhdan–Lusztig polynomials, combinatorial methods in representation theory (Kyoto, 1998). Adv. Stud. Pure Math. 28, 155–220 (2000)
Novelli, J.-C., Schilling, A.: The forgotten monoid, combinatorial representation theory and related topics. Res. Inst. Math. Sci. B8, 71–83 (2008)
Roberts, A.: Dual equivalence graphs revisited and the explicit Schur expansion of a family of LLT polynomials. J. Algebr. Comb. 39, 389–428 (2014)
Rosales, J.C., García-Sánchez, P.A.: Finitely Generated Commutative Monoids. Nova Science Publishers Inc, Commack (1999)
Schützenberger, M.-P.: La correspondance de Robinson. In: Combinatoire et représentation du groupe symétrique, Lecture Notes in Mathematics, vol. 579, pp. 59–113. Springer, Berlin (1977)
Stanley, R.P.: Enumerative Combinatorics, vol. 2. Cambridge University Press, Cambridge (1999)
Stembridge, J.R.: Admissible \(W\)-graphs represent. Theory 12, 346–368 (2008)
van Leeuwen, M.A.A.: Some objective correspondences involving domino tableaux. Electron. J. Comb. 7, 35 (2000)
Wachs, M.L.: Flagged Schur functions, Schubert polynomials, and symmetrizing operators. J. Comb. Theory Ser. A 40, 276–289 (1985)
Zabrocki, M.: Positivity for special cases of \((q, t)\)-Kostka coefficients and standard tableaux statistics. Electron. J. Comb. 6, 41 (1999)
Acknowledgments
We thank Sami Assaf, Anna Blasiak, Thomas Lam, and Bernard Leclerc for helpful discussions, and Elaine So and Xun Zhu for help typing and typesetting figures. This project began while the first author was a postdoc at the University of Michigan. He is grateful to John Stembridge for his generous advice and many detailed discussions.
Author information
Authors and Affiliations
Corresponding author
Additional information
The authors were supported by NSF Grants DMS-14071174 (J.B.) and DMS-1361789 (S.F.).
Rights and permissions
About this article
Cite this article
Blasiak, J., Fomin, S. Noncommutative Schur functions, switchboards, and Schur positivity. Sel. Math. New Ser. 23, 727–766 (2017). https://doi.org/10.1007/s00029-016-0253-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00029-016-0253-y