Combinatorial Formulas for Macdonald and Hall-Littlewood Polynomials

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.

Keywords

• macdonald polynomials
• hall-littlewood polynomials
• haglund-haiman-loehr formula
• alcove walks
• ram-yip formula
• schwer's formula
• [math.math-co] mathematics [math]/combinatorics [math.co]
• [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]