Noncommutative chromatic quasi-symmetric functions, Macdonald polynomials, and the Yang-Baxter equation

TL;DR

This paper develops a noncommutative analogue of Macdonald polynomials linked to the Yang-Baxter equation, extending chromatic quasi-symmetric functions into noncommuting variables and exploring their algebraic and combinatorial properties.

Contribution

It introduces a noncommutative version of Macdonald polynomials compatible with a noncommutative Haglund-Wilson formula and conjectures their relation to Yang-Baxter elements of Hecke algebras.

Findings

Noncommutative Macdonald polynomials for rectangular partitions match multi-t Hall-Littlewood functions at q=0.

Conjecture: multi-t Macdonald polynomials are traces of Yang-Baxter elements in Hecke algebras.

Relations between quasi-symmetric chromatic functions and Hecke algebra calculations are established.

Abstract

As shown in our paper [JCTA 177 (2021), Paper No. 105305], the chromatic quasi-symmetric function of Shareshian-Wachs can be lifted to ${\bf WQSym}$, the algebra of quasi-symmetric functions in noncommuting variables. We investigate here its behaviour with respect to classical transformations of alphabets and propose a noncommutative analogue of Macdonald polynomials compatible with a noncommutative version of the Haglund-Wilson formula. We also introduce a multi-$t$ version of these noncommutative analogues. For rectangular partitions, their commutative images at $q=0$ appear to coincide with the multi-$t$ Hall-Littlewood functions introduced in [Lett. Math. Phys. 35 (1995), 359]. This leads us to conjecture that for rectangular partitions, multi-$t$ Macdonald polynomials are obtained as equivariant traces of certain Yang-Baxter elements of Hecke algebras. We also conjecture that all (ordinary) Macdonald polynomials can be obtained in this way. We conclude with some remarks relating various aspects of quasi-symmetric chromatic functions to calculations in Hecke algebras. In particular, we show that all modular relations are given by the product formula of the Kazhdan-Lusztig basis.