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Journal of the American Mathematical Society
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
Published electronically: April 8, 2005
MathSciNet review: 2138143
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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 $\leq 2$.


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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: http://dx.doi.org/10.1090/S0894-0347-05-00485-6
PII: S 0894-0347(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.