Modified Macdonald polynomials and the multispecies zero-range process: I

Ayyer, Arvind; Mandelshtam, Olya; Martin, James B

doi:10.5802/alco.248

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

Algebraic Combinatorics, Volume 6 (2023) no. 1, pp. 243-284.

Abstract

In this paper we prove a new combinatorial formula for the modified Macdonald polynomials ${\tilde{H}}_{λ} (X; q, t)$ , 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 $P_{λ} (X; q, t)$ . In the case $q = 1$ and $X = (1, 1, \dots, 1)$ , 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 $X = (x_{1}, \dots, x_{n})$ ; the $x_{i}$ play the role of site-dependent jump rates for the mTAZRP.

Received: 2021-08-09
Revised: 2022-03-15
Accepted: 2022-06-14
Published online: 2023-02-24

DOI: 10.5802/alco.248

Classification: 05E05, 05A10, 05A19, 05A05, 33D52
Keywords: modified Macdonald polynomials, TAZRP, tableaux, zero range process

Author's affiliations:

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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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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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