Combinatorial formulas for symmetric Macdonald polynomials by superizations

TL;DR

This work advances the combinatorial understanding of Macdonald polynomials by developing new formulas for $P_{\lambda}(X;q,t)$ and $J_{\lambda}(X;q,t)$ through a refined superization framework. Central to the approach are the quadruple coinversion statistic $\overline{\mathsf{quadcoinv}}$ and its complements, together with flip operators and positive distinguished subexpressions that align major index with these statistics across non-attacking fillings. The authors establish key equidistribution results, derive explicit sum-product formulas, and recover multiple existing Macdonald formulas as special cases, including Lenart’s distinct-part case, via unified combinatorial machinery. They also present alternative proofs (via coinv$'$ and mixinv perspectives) and outline a broader sixteen-statistic landscape $\mathcal{A}$, illustrating deep connections to modified Macdonald polynomials, multiline queues, and qKZ theory. The results offer a unified, flexible toolkit for Macdonald combinatorics with potential implications for representation theory and algebraic geometry.

Abstract

In this paper, we derive new combinatorial formulas for symmetric Macdonald polynomials $P_λ(X;q,t)$ and integral Macdonald polynomials $J_λ(X;q,t)$, in terms of several new statistics and the major index for a partition $λ$. Moreover, three existing formulas for symmetric Macdonald polynomials established by Corteel--Mandelshtam--Williams (2022), Corteel--Haglund--Mandelshtam--Mason--Williams (2022) and Mandelshtam (2025) are recovered. Our proof relies on a new statistic on super fillings, employing the superization formula of Ayyer--Mandelshtam--Martin (2023) and our recent approach to modified Macdonald polynomials.

Figures (11)

• Figure 2.1: The Young diagram $\mathsf{dg}(\lambda)$ of $\lambda=(4,4,2,2,1)$ and its conjugate Young diagram $\mathsf{dg}'(\lambda)$ from left to right.
• Figure 2.2: A quadruple $(z,w,u,v)$ and a queue inversion triple $(z,u,v)$
• Figure 5.1: Subcases (i) and (ii) for inseparable super fillings
• Figure 8.1: A diagram with boxes labelled by its reading order.
• Figure : queue inversion triple

Theorems & Definitions (63)

• Definition 1.1: Symmetric Macdonald polynomials
• Example 1.1
• Example 1.2
• Definition 1.2: Modified Macdonald polynomials
• Example 1.3
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5