# ALGEBRAIC COMBINATORICS

Modified Macdonald polynomials and the multispecies zero-range process: I
Algebraic Combinatorics, Volume 6 (2023) no. 1, pp. 243-284.

In this paper we prove a new combinatorial formula for the modified Macdonald polynomials ${\stackrel{˜}{H}}_{\lambda }\left(X;q,t\right)$, motivated by connections to the theory of interacting particle systems from statistical mechanics. The formula involves a new statistic called queue inversions on fillings of tableaux. This statistic is closely related to the multiline queues which were recently used to give a formula for the Macdonald polynomials ${P}_{\lambda }\left(X;q,t\right)$. In the case $q=1$ and $X=\left(1,1,\cdots ,1\right)$, that formula had also been shown to compute stationary probabilities for a particle system known as the multispecies ASEP on a ring, and it is natural to ask whether a similar connection exists between the modified Macdonald polynomials and a suitable statistical mechanics model. In a sequel to this work, we demonstrate such a connection, showing that the stationary probabilities of the multispecies totally asymmetric zero-range process (mTAZRP) on a ring can be computed using tableaux formulas with the queue inversion statistic. This connection extends to arbitrary $X=\left({x}_{1},\cdots ,{x}_{n}\right)$; the ${x}_{i}$ play the role of site-dependent jump rates for the mTAZRP.

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.248
Classification: 05E05, 05A10, 05A19, 05A05, 33D52
Keywords: modified Macdonald polynomials, TAZRP, tableaux, zero range process
Ayyer, Arvind 1; Mandelshtam, Olya 2; Martin, James B 3

1 Department of Mathematics Indian Institute of Science Bangalore 560 012, India
2 Department of Combinatorics and Optimization University of Waterloo Waterloo, ON, Canada
3 Department of Statistics University of Oxford UK
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Ayyer, Arvind; Mandelshtam, Olya; Martin, James B. Modified Macdonald polynomials and the multispecies zero-range process: I. Algebraic Combinatorics, Volume 6 (2023) no. 1, pp. 243-284. doi : 10.5802/alco.248. https://alco.centre-mersenne.org/articles/10.5802/alco.248/

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