Macdonald polynomials at t = 0 through twisted multiline queues
Abstract
Multiline queues are versatile combinatorial objects that play a key role in understanding the remarkable connection between the asymmetric simple exclusion process (ASEP) on a circle and Macdonald polynomials. Specializing the results of Corteel--Mandelshtam--Williams (2018) to the case yields a formula for the -Whittaker polynomials through the Ferrari--Martin (2007) algorithm with a major index () statistic. In this paper, we reinterpret the statistic as a statistic on reading words, thereby bypassing the Ferrari--Martin algorithm to obtain an elegant formula for the -Whittaker polynomials. Our methods naturally extend to the case of bosonic multiline queues, with which we obtain analogous results for the modified Hall--Littlewood polynomials using a statistic on reading words. Twisted multiline queues (GMLQs) are obtained from the action of the symmetric group on the rows of a multiline queue. The Ferrari--Martin algorithm was extended to GMLQs by Arita--Ayyer--Mallick--Prolhac (2011), and Aas--Grinberg--Scrimshaw (2020) showed it is preserved under this action. We extend these results by defining a statistic on GMLQs that is also preserved under this action. This yields a novel family of formulas, indexed by compositions, for the -Whittaker polynomials. Additionally, we define a procedure on both GMLQs and bosonic multiline queues that we call collapsing, which can can be realized via the Kashiwara (crystal) operators on type-A Kirillov--Reshetikhin crystals. As an application, we naturally recover the Lascoux--Sch\"utzenberger formula for the -Whittaker and modified Hall--Littlewood polynomials, and the classical and dual Cauchy identities for Schur functions.
Cite
@article{arxiv.2407.05362,
title = {Macdonald polynomials at t = 0 through twisted multiline queues},
author = {Olya Mandelshtam and Jerónimo Valencia-Porras},
journal= {arXiv preprint arXiv:2407.05362},
year = {2025}
}