Combinatorial formulas for Macdonald polynomials by superizations

Abstract

In this paper, we derive new combinatorial formulas for symmetric Macdonald polynomials $P_{\hat{I}}»(X;q,t)$ and non-symmetric Macdonald polynomials $E_{\hat{I}}^3(X;q,t)$, in terms of several new statistics and the major index, for a partition $\hat{I}»$ and a weak composition ${\hat{I}}^3$. Compared to previous formulas, these new formulas contain the fewest terms and lead to explicit $(q,t)$-formulas for the coefficients in the monomial expansion of $P_{\hat{I}}»(X;q,t)$. In particular, the combinatorial formula for $E_{\hat{I}}^3(X;q,t)$ extends the one for $E_{\hat{I}}»(X;q,t)$ indexed by a partition $\hat{I}»$, due to Corteel, Mandelshtam and Williams (2022). Three existing formulas for $P_{\hat{I}}»(X;q,t)$ established by Corteel, Mandelshtam and Williams (2022), by Corteel, Haglund, Mandelshtam, Mason and Williams (2022), and by Mandelshtam (2025) are recovered. Our proof relies on two new statistics on super fillings, employing the superization formulas of Haglund--Haiman--Loehr (2005) and Ayyer--Mandelshtam--Martin (2023), together with our recent approach to modified Macdonald polynomials.

47 pages, 16 figures; The (q,t)-formula for symmetric Macdonald Polynomials and the combinatorial formula for non-symmetric Macdonald Polynomials are added