A combinatorial formula for Macdonald polynomials

Haglund, J. ¹   ; Haiman, M. ²   ; Loehr, N. ^{1 ,  3}

¹ Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395
² Department of Mathematics, University of California, Berkeley, California 97420-3840
³ Department of Mathematics, College of William & Mary, Williamsburg, Virginia 23187-8795

Journal of the American Mathematical Society, Tome 18 (2005) no. 3, pp. 735-761

Résumé

We prove a combinatorial formula for the Macdonald polynomial $\tilde {H}_{\mu }(x;q,t)$ which had been conjectured by Haglund. Corollaries to our main theorem include the expansion of $\tilde {H}_{\mu }(x;q,t)$ in terms of LLT polynomials, a new proof of the charge formula of Lascoux and Schützenberger for Hall-Littlewood polynomials, a new proof of Knop and Sahi’s combinatorial formula for Jack polynomials as well as a lifting of their formula to integral form Macdonald polynomials, and a new combinatorial rule for the Kostka-Macdonald coefficients $\tilde {K}_{\lambda \mu }(q,t)$ in the case that $\mu$ is a partition with parts $\leq 2$.

Détail

DOI : 10.1090/S0894-0347-05-00485-6

Bibliographie

[1] Butler, Lynne M. Subgroup lattices and symmetric functions Mem. Amer. Math. Soc. 1994

[2] Carré, Christophe, Leclerc, Bernard Splitting the square of a Schur function into its symmetric and antisymmetric parts J. Algebraic Combin. 1995 201 231

[3] Fishel, Susanna Statistics for special 𝑞,𝑡-Kostka polynomials Proc. Amer. Math. Soc. 1995 2961 2969

[4] Kirillov, Anatol N., Noumi, Masatoshi Affine Hecke algebras and raising operators for Macdonald polynomials Duke Math. J. 1998 1 39

[5] Kirillov, Anatol N., Noumi, Masatoshi Affine Hecke algebras and raising operators for Macdonald polynomials Duke Math. J. 1998 1 39

[6] Garsia, A. M., Zabrocki, M. Polynomiality of the 𝑞,𝑡-Kostka revisited 2001 473 491

[7] Haglund, J. A combinatorial model for the Macdonald polynomials Proc. Natl. Acad. Sci. USA 2004 16127 16131

[8] Haglund, J., Haiman, M., Loehr, N., Remmel, J. B., Ulyanov, A. A combinatorial formula for the character of the diagonal coinvariants Duke Math. J. 2005 195 232

[9] Haiman, Mark Macdonald polynomials and geometry 1999 207 254

[10] Haiman, Mark Hilbert schemes, polygraphs and the Macdonald positivity conjecture J. Amer. Math. Soc. 2001 941 1006

[11] Haiman, Mark Notes on Macdonald polynomials and the geometry of Hilbert schemes 2002 1 64

[12] Kashiwara, M. On crystal bases of the 𝑄-analogue of universal enveloping algebras Duke Math. J. 1991 465 516

[13] Kashiwara, M., Miwa, T., Stern, E. Decomposition of 𝑞-deformed Fock spaces Selecta Math. (N.S.) 1995 787 805

[14] Kirillov, Anatol N., Noumi, Masatoshi Affine Hecke algebras and raising operators for Macdonald polynomials Duke Math. J. 1998 1 39

[15] Kirillov, Anatol N., Noumi, Masatoshi Affine Hecke algebras and raising operators for Macdonald polynomials Duke Math. J. 1998 1 39

[16] Knop, Friedrich, Sahi, Siddhartha A recursion and a combinatorial formula for Jack polynomials Invent. Math. 1997 9 22

[17] Lapointe, L., Morse, J. Tableaux statistics for two part Macdonald polynomials 2003 61 84

[18] Lapointe, Luc, Vinet, Luc Rodrigues formulas for the Macdonald polynomials Adv. Math. 1997 261 279

[19] Lascoux, Alain, Leclerc, Bernard, Thibon, Jean-Yves Ribbon tableaux, Hall-Littlewood functions, quantum affine algebras, and unipotent varieties J. Math. Phys. 1997 1041 1068

[20] Lascoux, Alain, Schützenberger, Marcel-Paul Sur une conjecture de H. O. Foulkes C. R. Acad. Sci. Paris Sér. A-B 1978

[21] Lascoux, Alain, Schützenberger, Marcel-Paul Croissance des polynômes de Foulkes-Green C. R. Acad. Sci. Paris Sér. A-B 1979

[22] Lascoux, Alain, Schützenberger, Marcel-P. Le monoïde plaxique 1981 129 156

[23] Leclerc, Bernard, Thibon, Jean-Yves Littlewood-Richardson coefficients and Kazhdan-Lusztig polynomials 2000 155 220

[24] Macdonald, I. G. Symmetric functions and Hall polynomials 1995

[25] Manivel, Laurent Symmetric functions, Schubert polynomials and degeneracy loci 2001

[26] Kirillov, Anatol N., Noumi, Masatoshi Affine Hecke algebras and raising operators for Macdonald polynomials Duke Math. J. 1998 1 39

[27] Schützenberger, M.-P. La correspondance de Robinson 1977 59 113

[28] Stanley, Richard P. Enumerative combinatorics. Vol. 2 1999

[29] Van Leeuwen, Marc A. A. Some bijective correspondences involving domino tableaux Electron. J. Combin. 2000

[30] Zabrocki, Mike Positivity for special cases of (𝑞,𝑡)-Kostka coefficients and standard tableaux statistics Electron. J. Combin. 1999