A combinatorial formula for Macdonald polynomials

TL;DR

This paper introduces a unified combinatorial formula for Macdonald polynomials across all Lie types using alcove walk combinatorics, generalizing previous specific formulas and connecting to various special cases.

Contribution

It provides a new uniform combinatorial formula for Macdonald polynomials applicable to all Lie types, extending prior type-specific results.

Findings

Unified combinatorial formula for Macdonald polynomials across all Lie types

Specializations recover known formulas for spherical functions and Weyl characters

Connects alcove walk combinatorics with Macdonald polynomial theory

Abstract

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 GL(n). 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-Littelmann).