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arXiv · 2407.05362

Macdonald polynomials at t = 0 through twisted multiline queues

Olya Mandelshtam·Jerónimo Valencia-Porras
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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 $t=0$ case yields a formula for the $q$-Whittaker polynomials through the Ferrari--Martin (2007) algorithm with a major index ($\texttt{maj}$) statistic. In this paper, we reinterpret the $\texttt{maj}$ statistic as a $\texttt{charge}$ statistic on reading words, thereby bypassing the Ferrari--Martin algorithm to obtain an elegant formula for the $q$-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 $\texttt{cocharge}$ 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 $\texttt{maj}$ statistic on GMLQs that is also preserved under this action. This yields a novel family of formulas, indexed by compositions, for the $q$-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 $\texttt{charge}$ formula for the $q$-Whittaker and modified Hall--Littlewood polynomials, and the classical and dual Cauchy identities for Schur functions.

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Olya Mandelshtam, Jerónimo Valencia-Porras. 2024-07-07. Macdonald polynomials at t = 0 through twisted multiline queues. https://arxiv.org/abs/2407.05362

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