Last update: 18 February 2013
In this paper we use the combinatorics of alcove walks to give a uniform combinatorial formula for Macdonald polynomials for all Lie types. These formulas are generalizations of the formulas of Haglund-Haiman-Loehr for Macdonald polynoimals of type At these formulas specialize to the formula of Schwer for the Macdonald spherical function in terms of positively folded alcove walks and at 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-Littelmann).
The Macdonald polynomials were introduced in the mid 1980s [Mac1992] [Mac0011046] as a remarkable family of orthogonal polynomials generalizing the spherical functions for a group, the Weyl characters, the Jack polynomials and the zonal polynomials. In the early 1990s Cherednik [Che1992] introduced the double affine Hecke algebra (the DAHA) and used it as a tool to prove conjectures of Macdonald. The DAHA is a fundamental tool for studying Macdonald polynomials. Using the DAHA, the nonsymmetric Macdonald polynomials Eμ can be constructed by applying products of “intertwining operators” to the generator of the polynomial representation of the DAHA (see [Hai2275709, Prop. 6.13]), and the symmetric Macdonald polynomials can then be constructed from the by “symmetrizing” (see [Mac1423624, Remarks after (6.8)]).
Of recent note in the theory of Macdonald polynomials has been the success of Haglund-Haiman-Loehr in giving, in the type case, explicit combinatorial formulas for the expansion of Macdonald polynomials in terms of monomials. These formulas were conjectured by J. Haglund and proved by Haglund-Haiman-Loehr in [HHL0409538] and [HHL0601693]. The papers [GRe2005] and [Hai2275709] are excellent survey articles discussing these developments.
Following a key idea of C. Schwer [Sch0506287], the paper [Ram0601343] developed a combinatorics for working in the affine Hecke algebra, the alcove walk model. It turns out that this combinatorics is the ideal tool for expansion of products of intertwining operators in the DAHA. These expansions, when applied to the generator of the polynomial representation of the DAHA, give formulas for the Macdonald polynomials which are generalizations, to all root systems, of the formulas obtained by Haglund-Haiman-Loehr [HHL0409538] [HHL0601693] in type
At the symmetric Macdonald polynomials are the Hall-Littlewood polynomials or the Macdonald spherical functions. These are the spherical functions for where is a group and is a maximal compact subgroup. The work of Schwer [Sch0506287, Thm. 1.1] provided fomulas for the expansion of the Macdonald spherical functions in terms of positively folded alcove walks. See [Ram0601343, Thm. 4.2(a)] for a description of the Schwer-KLM formula in terms of the alcove walk model. The formula for Macdonald polynomials which we give in Theorem 3.4 reduces to the Schwer formula at
At the symmetric Macdonald polynomials are the Weyl characters or Schur functions. In this case our formula for the Macdonald polynomial specializes to the formula for the Weyl character in terms of the Littelmann path model (in the maximal dimensional positively folded gallery form of Gaussent-Littelmann [GLi2002, Cor. 1 p. 62]).
It is interesting to note that, in the formulas for the symmetric Macdonald polynomials, the negative folds and the positive folds play an equal role. It is known [GLi2002] that the alcove walks with only positive folds contain detailed information about the geometry of Mirković-Vilonen intersections in the loop Grassmannian. It is tantalizing to wonder whether the alcove walks with both positive and negative folds play a similar role in the geometry of flag varieties for reductive groups over two dimensional local fields and whether the expansions of Macdonald polynomials in this paper are shadows of geometric decompositions.
The papers [GLi2002] and [Ram0601343] explain how the combinatorics of alcove walks is almost equivalent to the combinatorics of crystal bases and Kashiwara operators (at least for the positively folded alcove walks of maximal dimension). Our expansions of Macdonald polynomials in terms of alcove walks give insight into possible relationships between Macdonald polynomials and crystal and canonical bases.
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 many instructive conversations about double affine Hecke algebras and Macdonald polynomials, without which this paper would never have been possible.
This is an excerpt from a paper entitled A combinatorial formula for Macdonald polynomials authored by Arun Ram and Martha Yip. It was dedicated to Adriano Garsia.