Combinatorial Formulas for Macdonald and Hall-Littlewood Polynomials of Types A and C. Extended Abstract

TL;DR

This paper connects the combinatorial formulas for Macdonald polynomials of types A and C, extending known results and providing a new approach for Hall-Littlewood polynomials of type C, which are special cases of Macdonald polynomials.

Contribution

It demonstrates that the Haglund-Haiman-Loehr formula can be derived from the Ram-Yip formula via compression and extends this approach to type C Hall-Littlewood polynomials.

Findings

Haglund-Haiman-Loehr formula follows from Ram-Yip formula

Extension of combinatorial formulas to type C Hall-Littlewood polynomials

First step towards formulas beyond type A

Abstract

A 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 fillings of Young diagrams. Recently, Ram and Yip gave a formula for the Macdonald polynomials of arbitrary type in terms of the corresponding affine Weyl group. In this paper, we show that a Haglund-Haiman-Loehr type formula follows naturally from the more general Ram-Yip formula, via compression. Then we extend this approach to the Hall-Littlewood polynomials of type C, which are specializations of the corresponding Macdonald polynomials at q=0. We note that no analog of the Haglund-Haiman-Loehr formula exists beyond type A, so our work is a first step towards finding such a formula.