Macdonald polynomials at t = 0 through twisted multiline queues

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 ($\mathtt{\text{maj}}$) statistic. In this paper, we reinterpret the $\mathtt{\text{maj}}$ statistic as a $\mathtt{\text{charge}}$ 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 $\mathtt{\text{cocharge}}$ 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 $\mathtt{\text{maj}}$ 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–Schützenberger $\mathtt{\text{charge}}$ formula for the $q$-Whittaker and modified Hall–Littlewood polynomials, and the classical and dual Cauchy identities for Schur functions.