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 μ = (μ1, . . ., μn). 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 Yi for all 1 6 i 6 n, and are thus equal to nonsymmetric Macdonald polynomials Eμ. 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 Eμ due to Haglund–Haiman–Loehr.