Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

A combinatorial formula for Macdonald polynomials


Authors: J. Haglund, M. Haiman and N. Loehr
Journal: J. Amer. Math. Soc. 18 (2005), 735-761
MSC (2000): Primary 05E10; Secondary 05A30
DOI: https://doi.org/10.1090/S0894-0347-05-00485-6
Published electronically: April 8, 2005
MathSciNet review: 2138143
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: 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 $\leq 2$.


References [Enhancements On Off] (What's this?)

  • 1. Lynne M. Butler, Combinatorial properties of partially ordered sets associated with partitions and finite abelian groups, Ph.D. thesis, Massachusetts Institute of Technology, 1986.
  • 2. -, Subgroup lattices and symmetric functions, Mem. Amer. Math. Soc. 112 (1994), no. 539, vi+160. MR 1223236 (95e:05122)
  • 3. Christophe Carré and Bernard Leclerc, Splitting the square of a Schur function into its symmetric and antisymmetric parts, J. Algebraic Combin. 4 (1995), no. 3, 201-231. MR 1331743 (97b:05165)
  • 4. Susanna Fishel, Statistics for special $q,t$-Kostka polynomials, Proc. Amer. Math. Soc. 123 (1995), no. 10, 2961-2969.MR 1264811 (95m:05244)
  • 5. A. M. Garsia and J. Remmel, Plethystic formulas and positivity for $q,t$-Kostka coefficients, Mathematical essays in honor of Gian-Carlo Rota (Cambridge, MA, 1996), Birkhäuser Boston, Boston, MA, 1998, pp. 245-262.MR 1627327 (99j:05189d)
  • 6. A. M. Garsia and G. Tesler, Plethystic formulas for Macdonald $q,t$-Kostka coefficients, Adv. Math. 123 (1996), no. 2, 144-222.MR 1420484 (99j:05189e)
  • 7. A. M. Garsia and M. Zabrocki, Polynomiality of the $q,t$-Kostka revisited, Algebraic combinatorics and computer science, Springer Italia, Milan, 2001, pp. 473-491, arXiv:math.QA/0008199. MR 1854489 (2002h:05160)
  • 8. J. Haglund, A combinatorial model for the Macdonald polynomials, Proc. Nat. Acad. Sci. U.S.A. 101 (2004), no. 46, 16127-16131.MR 2114585
  • 9. J. Haglund, M. Haiman, N. Loehr, J. B. Remmel, and A. Ulyanov, A combinatorial formula for the character of the diagonal coinvariants, Duke Math. J. 126 (2005), no. 2, 195-232, arXiv:math.CO/0310424.MR 2115257
  • 10. Mark Haiman, Macdonald polynomials and geometry, New perspectives in geometric combinatorics (Billera, Björner, Greene, Simion, and Stanley, eds.), MSRI Publications, vol. 38, Cambridge University Press, 1999, pp. 207-254.MR 1731818 (2001k:05203)
  • 11. -, Hilbert schemes, polygraphs and the Macdonald positivity conjecture, J. Amer. Math. Soc. 14 (2001), no. 4, 941-1006, arXiv:math.AG/0010246.MR 1839919 (2002c:14008)
  • 12. -, Notes on Macdonald polynomials and the geometry of Hilbert schemes, Symmetric functions 2001: surveys of developments and perspectives, NATO Sci. Ser. II Math. Phys. Chem., vol. 74, Kluwer Acad. Publ., Dordrecht, 2002, pp. 1-64.MR 2059359
  • 13. M. Kashiwara, On crystal bases of the $Q$-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), no. 2, 465-516.MR 1115118 (93b:17045)
  • 14. M. Kashiwara, T. Miwa, and E. Stern, Decomposition of $q$-deformed Fock spaces, Selecta Math. (N.S.) 1 (1995), no. 4, 787-805, arXiv:q-alg/9508006.MR 1383585 (97c:17021)
  • 15. Anatol N. Kirillov and Masatoshi Noumi, Affine Hecke algebras and raising operators for Macdonald polynomials, Duke Math. J. 93 (1998), no. 1, 1-39, arXiv:q-alg/9605004.MR 1620075 (99j:05189a)
  • 16. Friedrich Knop, Integrality of two variable Kostka functions, J. Reine Angew. Math. 482 (1997), 177-189, arXiv:q-alg/9603027.MR 1427661 (99j:05189c)
  • 17. Friedrich Knop and Siddhartha Sahi, A recursion and a combinatorial formula for Jack polynomials, Invent. Math. 128 (1997), no. 1, 9-22, arXiv:q-alg/9610016.MR 1437493 (98k:33040)
  • 18. L. Lapointe and J. Morse, Tableaux statistics for two part Macdonald polynomials, Algebraic combinatorics and quantum groups, World Sci. Publishing, River Edge, NJ, 2003, pp. 61-84, arXiv:math.CO/9812001.MR 2035130
  • 19. Luc Lapointe and Luc Vinet, Rodrigues formulas for the Macdonald polynomials, Adv. Math. 130 (1997), no. 2, 261-279, arXiv:q-alg/9607025.MR 1472319 (99e:05128)
  • 20. Alain Lascoux, Bernard Leclerc, and Jean-Yves Thibon, Ribbon tableaux, Hall-Littlewood functions, quantum affine algebras, and unipotent varieties, J. Math. Phys. 38 (1997), no. 2, 1041-1068, arXiv:q-alg/9512031.MR 1434225 (98c:05167)
  • 21. Alain Lascoux and Marcel-Paul Schützenberger, Sur une conjecture de H. O. Foulkes, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 7, A323-A324.MR 0472993 (57:12672)
  • 22. -, Croissance des polynômes de Foulkes-Green, C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 2, A95-A98.MR 0524758 (80d:20052)
  • 23. -, Le monoïde plaxique, Noncommutative structures in algebra and geometric combinatorics (Naples, 1978), CNR, Rome, 1981, pp. 129-156.MR 0646486 (83g:20016)
  • 24. Bernard Leclerc and Jean-Yves Thibon, Littlewood-Richardson coefficients and Kazhdan-Lusztig polynomials, Combinatorial methods in representation theory (Kyoto, 1998), Adv. Stud. Pure Math., vol. 28, Kinokuniya, Tokyo, 2000, pp. 155-220, arXiv:math.QA/9809122.MR 1864481 (2002k:20014)
  • 25. I. G. Macdonald, A new class of symmetric functions, Actes du 20e Séminaire Lotharingien, vol. 372/S-20, Publications I.R.M.A., Strasbourg, 1988, pp. 131-171.
  • 26. -, Symmetric functions and Hall polynomials, second ed., The Clarendon Press, Oxford University Press, New York, 1995, with contributions by A. Zelevinsky, Oxford Science Publications. MR 1354144 (96h:05207)
  • 27. Laurent Manivel, Symmetric functions, Schubert polynomials and degeneracy loci, SMF/AMS Texts and Monographs, vol. 6, American Mathematical Society, Providence, RI, 2001, Translated from the 1998 French original by John R. Swallow, Cours Spécialisés [Specialized Courses], 3.MR 1852463 (2002h:05161)
  • 28. Siddhartha Sahi, Interpolation, integrality, and a generalization of Macdonald's polynomials, Internat. Math. Res. Notices 10 (1996), 457-471.MR 1399411 (99j:05189b)
  • 29. M.-P. Schützenberger, La correspondance de Robinson, Combinatoire et représentation du groupe symétrique (Actes Table Ronde CNRS, Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976), Springer, Berlin, 1977, Lecture Notes in Mathematics, Vol. 579, pp. 59-113. MR 0498826 (58:16863)
  • 30. Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge University Press, Cambridge, 1999, With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. MR 1676282 (2000k:05026)
  • 31. Marc A. A. van Leeuwen, Some bijective correspondences involving domino tableaux, Electron. J. Combin. 7 (2000), Research Paper 35, 25 pp. (electronic), arXiv:math.CO/9909119. MR 1769066 (2001d:05195)
  • 32. Mike Zabrocki, Positivity for special cases of $(q,t)$-Kostka coefficients and standard tableaux statistics, Electron. J. Combin. 6 (1999), Research Paper 41, 36 pp. (electronic), arXiv:math.CO/9901016.MR 1725705 (2001g:05101)

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 05E10, 05A30

Retrieve articles in all journals with MSC (2000): 05E10, 05A30


Additional Information

J. Haglund
Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395
Email: jhaglund@math.upenn.edu

M. Haiman
Affiliation: Department of Mathematics, University of California, Berkeley, California 97420-3840

N. Loehr
Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395
Address at time of publication: Department of Mathematics, College of William & Mary, Williamsburg, Virginia 23187-8795
Email: nloehr@math.upenn.edu, nick@math.wm.edu

DOI: https://doi.org/10.1090/S0894-0347-05-00485-6
Received by editor(s): October 18, 2004
Published electronically: April 8, 2005
Additional Notes: The first author’s work was supported by NSA grant MSPF-02G-193
The second author’s work was supported by NSF grant DMS-0301072
The third author’s work was supported by an NSF Postdoctoral Research Fellowship
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society