A combinatorial formula for Macdonald polynomials

Arun Ram, Martha Yip

Research output: Contribution to journalArticlepeer-review

46 Scopus citations


In this paper we use the combinatorics of alcove walks to give uniform combinatorial formulas for Macdonald polynomials for all Lie types. These formulas resemble the formulas of Haglund, Haiman and Loehr for Macdonald polynomials of type GLn. At q=0 these formulas specialize to the formula of Schwer for the Macdonald spherical function in terms of positively folded alcove walks and at q=t=0 these formulas specialize to the formula for the Weyl character in terms of the Littelmann path model (in the positively folded gallery form of Gaussent and Littelmann).

Original languageEnglish
Pages (from-to)309-331
Number of pages23
JournalAdvances in Mathematics
Issue number1
StatePublished - Jan 15 2011

Bibliographical note

Funding Information:
This research was partially supported by the National Science Foundation (NSF) under grant DMS-0353038 at the University of Wisconsin, Madison. We thank the NSF for continuing support of our research. This paper was completed while the authors were in residence at the special semester in Combinatorial Representation Theory at Mathematical Sciences Research Institute (MSRI). It is a pleasure to thank MSRI for hospitality, support and a wonderful and stimulating working environment. A. Ram thanks S. Griffeth for many instructive conversations about double affine Hecke algebras and Macdonald polynomials.


  • Alcove walks
  • Combinatorial formulas
  • Macdonald polynomials
  • Path model
  • Symmetric functions

ASJC Scopus subject areas

  • General Mathematics


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