The Haglund--Haiman--Loehr theorem (2005) provides a combinatorial formula for the modified Macdonald polynomials, highlighting the surprising connections between modified Macdonald polynomials and combinatorial statistics such as the major index and the inversion number on fillings of Young diagrams.
Inspired by Martin's multiline-queue formula (2020) for the stationary distribution of multi-type asymmetric simple exclusion processes, Corteel, Haglund, Mandelshtam, Mason and Williams (2021) introduced the queue inversion statistic quinv and conjectured that the tableaux formula for the modified Macdonald polynomials is invariant if the inversion statistic inv is replaced by quinv.
This was subsequently resolved by Ayyer, Mandelshtam and Martin (2023), who proposed a stronger conjecture on the equivalence of the two refined formulas for modified Macdonald polynomials.
Our main result confirms this Ayyer--Mandelshtam--Martin conjecture. Beyond that, we discover a family of 16 statistics on fillings of any given Young diagram and prove new combinatorial formulas for modified Macdonald polynomials. Building upon these new formulas, we establish four compact formulas for modified Macdonald polynomials, which enable us to derive four explicit expressions for the monomial expansion of modified Macdonald polynomials, one of which coincides with the formula given by Garbali and Wheeler (2020). This talk is based on joint work with Xiaowei Lin.