## A combinatorial formula for Macdonald polynomials

HTML articles powered by AMS MathViewer

- by J. Haglund, M. Haiman and N. Loehr PDF
- J. Amer. Math. Soc.
**18**(2005), 735-761 Request permission

## 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 SchĂĽtzenberger for Hall-Littlewood polynomials, a new proof of Knop and Sahiâ€™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

- Butler86 Lynne M. Butler,
- Lynne M. Butler,
*Subgroup lattices and symmetric functions*, Mem. Amer. Math. Soc.**112**(1994), no.Â 539, vi+160. MR**1223236**, DOI 10.1090/memo/0539 - Christophe CarrĂ© and Bernard Leclerc,
*Splitting the square of a Schur function into its symmetric and antisymmetric parts*, J. Algebraic Combin.**4**(1995), no.Â 3, 201â€“231. MR**1331743**, DOI 10.1023/A:1022475927626 - Susanna Fishel,
*Statistics for special $q,t$-Kostka polynomials*, Proc. Amer. Math. Soc.**123**(1995), no.Â 10, 2961â€“2969. MR**1264811**, DOI 10.1090/S0002-9939-1995-1264811-3 - Anatol N. Kirillov and Masatoshi Noumi,
*Affine Hecke algebras and raising operators for Macdonald polynomials*, Duke Math. J.**93**(1998), no.Â 1, 1â€“39. MR**1620075**, DOI 10.1215/S0012-7094-98-09301-2 - Anatol N. Kirillov and Masatoshi Noumi,
*Affine Hecke algebras and raising operators for Macdonald polynomials*, Duke Math. J.**93**(1998), no.Â 1, 1â€“39. MR**1620075**, DOI 10.1215/S0012-7094-98-09301-2 - A. M. Garsia and M. Zabrocki,
*Polynomiality of the $q,t$-Kostka revisited*, Algebraic combinatorics and computer science, Springer Italia, Milan, 2001, pp.Â 473â€“491. MR**1854489** - J. Haglund,
*A combinatorial model for the Macdonald polynomials*, Proc. Natl. Acad. Sci. USA**101**(2004), no.Â 46, 16127â€“16131. MR**2114585**, DOI 10.1073/pnas.0405567101 - 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.Â 2, 195â€“232. MR**2115257**, DOI 10.1215/S0012-7094-04-12621-1 - Mark Haiman,
*Macdonald polynomials and geometry*, New perspectives in algebraic combinatorics (Berkeley, CA, 1996â€“97) Math. Sci. Res. Inst. Publ., vol. 38, Cambridge Univ. Press, Cambridge, 1999, pp.Â 207â€“254. MR**1731818** - Mark Haiman,
*Hilbert schemes, polygraphs and the Macdonald positivity conjecture*, J. Amer. Math. Soc.**14**(2001), no.Â 4, 941â€“1006. MR**1839919**, DOI 10.1090/S0894-0347-01-00373-3 - Mark Haiman,
*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.Â 1â€“64. MR**2059359**, DOI 10.1007/978-94-010-0524-1_{1} - M. Kashiwara,
*On crystal bases of the $Q$-analogue of universal enveloping algebras*, Duke Math. J.**63**(1991), no.Â 2, 465â€“516. MR**1115118**, DOI 10.1215/S0012-7094-91-06321-0 - M. Kashiwara, T. Miwa, and E. Stern,
*Decomposition of $q$-deformed Fock spaces*, Selecta Math. (N.S.)**1**(1995), no.Â 4, 787â€“805. MR**1383585**, DOI 10.1007/BF01587910 - Anatol N. Kirillov and Masatoshi Noumi,
*Affine Hecke algebras and raising operators for Macdonald polynomials*, Duke Math. J.**93**(1998), no.Â 1, 1â€“39. MR**1620075**, DOI 10.1215/S0012-7094-98-09301-2 - Anatol N. Kirillov and Masatoshi Noumi,
*Affine Hecke algebras and raising operators for Macdonald polynomials*, Duke Math. J.**93**(1998), no.Â 1, 1â€“39. MR**1620075**, DOI 10.1215/S0012-7094-98-09301-2 - Friedrich Knop and Siddhartha Sahi,
*A recursion and a combinatorial formula for Jack polynomials*, Invent. Math.**128**(1997), no.Â 1, 9â€“22. MR**1437493**, DOI 10.1007/s002220050134 - L. Lapointe and J. Morse,
*Tableaux statistics for two part Macdonald polynomials*, Algebraic combinatorics and quantum groups, World Sci. Publ., River Edge, NJ, 2003, pp.Â 61â€“84. MR**2035130**, DOI 10.1142/9789812775405_{0}004 - Luc Lapointe and Luc Vinet,
*Rodrigues formulas for the Macdonald polynomials*, Adv. Math.**130**(1997), no.Â 2, 261â€“279. MR**1472319**, DOI 10.1006/aima.1997.1662 - 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.Â 2, 1041â€“1068. MR**1434225**, DOI 10.1063/1.531807 - Alain Lascoux and Marcel-Paul SchĂĽtzenberger,
*Sur une conjecture de H. O. Foulkes*, C. R. Acad. Sci. Paris SĂ©r. A-B**286**(1978), no.Â 7, A323â€“A324 (French, with English summary). MR**472993** - Alain Lascoux and Marcel-Paul SchĂĽtzenberger,
*Croissance des polynĂ´mes de Foulkes-Green*, C. R. Acad. Sci. Paris SĂ©r. A-B**288**(1979), no.Â 2, A95â€“A98 (French, with English summary). MR**524758** - Alain Lascoux and Marcel-P. SchĂĽtzenberger,
*Le monoĂŻde plaxique*, Noncommutative structures in algebra and geometric combinatorics (Naples, 1978) Quad. â€śRicerca Sci.â€ť, vol. 109, CNR, Rome, 1981, pp.Â 129â€“156 (French, with Italian summary). MR**646486** - 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.Â 155â€“220. MR**1864481**, DOI 10.2969/aspm/02810155
Mac88 I. G. Macdonald, - I. G. Macdonald,
*Symmetric functions and Hall polynomials*, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR**1354144** - Laurent Manivel,
*Symmetric functions, Schubert polynomials and degeneracy loci*, SMF/AMS Texts and Monographs, vol. 6, American Mathematical Society, Providence, RI; SociĂ©tĂ© MathĂ©matique de France, Paris, 2001. Translated from the 1998 French original by John R. Swallow; Cours SpĂ©cialisĂ©s [Specialized Courses], 3. MR**1852463** - Anatol N. Kirillov and Masatoshi Noumi,
*Affine Hecke algebras and raising operators for Macdonald polynomials*, Duke Math. J.**93**(1998), no.Â 1, 1â€“39. MR**1620075**, DOI 10.1215/S0012-7094-98-09301-2 - M.-P. SchĂĽtzenberger,
*La correspondance de Robinson*, Combinatoire et reprĂ©sentation du groupe symĂ©trique (Actes Table Ronde CNRS, Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976) Lecture Notes in Math., Vol. 579, Springer, Berlin, 1977, pp.Â 59â€“113 (French). MR**0498826** - Richard P. Stanley,
*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 Fomin. MR**1676282**, DOI 10.1017/CBO9780511609589 - Marc A. A. van Leeuwen,
*Some bijective correspondences involving domino tableaux*, Electron. J. Combin.**7**(2000), Research Paper 35, 25. MR**1769066** - Mike Zabrocki,
*Positivity for special cases of $(q,t)$-Kostka coefficients and standard tableaux statistics*, Electron. J. Combin.**6**(1999), Research Paper 41, 36. MR**1725705**

*Combinatorial properties of partially ordered sets associated with partitions and finite abelian groups*, Ph.D. thesis, Massachusetts Institute of Technology, 1986.

*A new class of symmetric functions*, Actes du 20e SĂ©minaire Lotharingien, vol. 372/S-20, Publications I.R.M.A., Strasbourg, 1988, pp. 131â€“171.

## Additional Information

**J. Haglund**- Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395
- MR Author ID: 600170
- 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
- Received by editor(s): October 18, 2004
- Published electronically: April 8, 2005
- Additional Notes: The first authorâ€™s work was supported by NSA grant MSPF-02G-193

The second authorâ€™s work was supported by NSF grant DMS-0301072

The third authorâ€™s work was supported by an NSF Postdoctoral Research Fellowship - © Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2138143