Hall-Littlewood Polynomials, Alcove Walks, and Fillings of Young Diagrams

TL;DR

This paper connects combinatorial formulas for Hall-Littlewood polynomials with alcove walk models, deriving a new formula for type A polynomials by relating existing formulas through a compression technique.

Contribution

It introduces a new combinatorial formula for type A Hall-Littlewood P-polynomials by linking Schwer's alcove walk formula with Haglund-Haiman-Loehr's fillings approach.

Findings

Derived a Haglund-Haiman-Loehr type formula for type A Hall-Littlewood P-polynomials.

Connected alcove walk models with Young diagram fillings.

Provided a compression procedure relating different formulas.

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. The inversion statistic, which is the more intricate one, suffices for specializing a closely related formula to one for the type A Hall-Littlewood Q-polynomials (spherical functions on p-adic groups). An apparently unrelated development, at the level of arbitrary finite root systems, led to Schwer's formula (rephrased and rederived by Ram) for the Hall-Littlewood P-polynomials of arbitrary type. The latter formula is in terms of so-called alcove walks, which 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 deriving a Haglund-Haiman-Loehr type formula for the Hall-Littlewood P-polynomials of type A from Ram's version of Schwer's formula via a "compression" procedure.