Universal divided difference operators and solvable lattice models

Abstract

We construct a new class of solvable lattice models whose partition functions are given by the largest family of type A polynomial divided difference operators that satisfy the braid relations. These multi-parameter operators were introduced by A. N. Kirillov and are vast generalizations of the classical divided difference operators. As a consequence, our lattice model gives a uniform method of computing various functions that arise in Schubert calculus and representation theory, including Schubert polynomials, key polynomials, and many of their generalizations. Along the way we resolve several of Kirillov's conjectures. In particular, our lattice model gives a proof of Kirillov's conjecture that the Hecke--Grothendieck polynomials have non-negative coefficients. We also search for an algebraic interpretation of our construction and identify two degenerations of the parameters for which our lattice model arises from the R-matrices of the standard representations of the quantum groups Uq(sl(1|n)) and Uq(sl(n|1)).