# Nonsymmetric Macdonald polynomials via integrable vertex models

## TL;DR

This paper constructs nonsymmetric Macdonald polynomials using integrable vertex models, connecting algebraic, combinatorial, and integrable systems approaches to provide explicit formulas and interpretations.

## Contribution

It introduces a new construction of nonsymmetric Macdonald polynomials via integrable vertex models and their partition functions, linking algebraic and combinatorial perspectives.

## Key findings

- Partition functions are eigenfunctions of Cherednik-Dunkl operators.
- Partition functions have a combinatorial interpretation as coloured lattice paths on a cylinder.
- Bijection recovers Haglund-Haiman-Loehr formula for Macdonald polynomials.

## Abstract

Starting from an integrable rank-$n$ vertex model, we construct an explicit family of partition functions indexed by compositions $\mu = (\mu_1,\dots,\mu_n)$. Using the Yang-Baxter algebra of the model and a certain rotation operation that acts on our partition functions, we show that they are eigenfunctions of the Cherednik-Dunkl operators $Y_i$ for all $1 \leq i \leq n$, and are thus equal to nonsymmetric Macdonald polynomials $E_{\mu}$. Our partition functions have the combinatorial interpretation of ensembles of coloured lattice paths which traverse a cylinder. Applying a simple bijection to such path ensembles, we show how to recover the well-known combinatorial formula for $E_{\mu}$ due to Haglund-Haiman-Loehr.