On Combinatorial Formulas for Macdonald Polynomials

TL;DR

This paper connects two major combinatorial formulas for Macdonald polynomials, showing how the Ram-Yip formula simplifies to a more concise version akin to Haglund-Haiman-Loehr's, reducing complexity.

Contribution

It demonstrates how the Ram-Yip formula can be compressed into a simpler form similar to the Haglund-Haiman-Loehr formula, unifying different approaches.

Findings

The Ram-Yip formula can be compressed to a more efficient form.

The new formula contains fewer terms than the original Ram-Yip formula.

The work unifies combinatorial formulas for Macdonald polynomials.

Abstract

A recent breakthrough in the theory of (type A) Macdonald polynomials is due to Haglund, Haiman and Loehr, who exhibited a combinatorial formula for these polynomials in terms of a pair of statistics on fillings of Young diagrams. Ram and Yip gave a formula for the Macdonald polynomials of arbitrary type in terms of so-called alcove walks; these originate in the work of Gaussent-Littelmann and of the author with Postnikov on discrete counterparts to the Littelmann path model. In this paper, we relate the above developments, by explaining how the Ram-Yip formula compresses to a new formula, which is similar to the Haglund-Haiman-Loehr one but contains considerably fewer terms.