Nonsymmetric Macdonald polynomials via integrable vertex models

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 $1 \leqslant i \leqslant n$, 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.