Elsevier

Advances in Mathematics

Volume 226, Issue 1, 15 January 2011, Pages 309-331
Advances in Mathematics

A combinatorial formula for Macdonald polynomials

https://doi.org/10.1016/j.aim.2010.06.022Get rights and content
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Abstract

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).

MSC

primary
05E05
secondary
33D52

Keywords

Macdonald polynomials
Symmetric functions
Path model
Alcove walks
Combinatorial formulas

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