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

Ayyer, Arvind¹ ; Mandelshtam, Olya² ; Martin, James B³

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, Tome 6 (2023) no. 1, pp. 243-284

Résumé

In this paper we prove a new combinatorial formula for the modified Macdonald polynomials ${\tilde{H}}_{\lambda }(X;q,t)$, 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 $P_{\lambda }(X;q,t)$. In the case $q=1$ and $X=(1,1,\cdots ,1)$, 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 $X=(x_1,\cdots ,x_n)$; the $x_i$ play the role of site-dependent jump rates for the mTAZRP.

Détail

Reçu le : 2021-08-09
Révisé le : 2022-03-15
Accepté le : 2022-06-14
Publié le : 2023-02-24

DOI : 10.5802/alco.248

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

Licence : CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits