Double Grothendieck polynomials and colored lattice models

TL;DR

This paper constructs integrable lattice models to interpret double Grothendieck polynomials, providing new proofs and connections to bumpless pipe dreams, flagged factorial Grothendieck polynomials, and determinant formulas for vexillary permutations.

Contribution

It introduces a colored six-vertex model for double Grothendieck polynomials and a semidual model for vexillary permutations, linking integrable systems with algebraic combinatorics.

Findings

Partition function equals double Grothendieck polynomial

New proof that stable limit is factorial Grothendieck polynomial

Determinant formula for double Schubert polynomials

Abstract

We construct an integrable colored six-vertex model whose partition function is a double Grothendieck polynomial. This gives an integrable systems interpretation of bumpless pipe dreams and recent results of Weigandt [arXiv:2003.07342] relating double Grothendieck polynomias with bumpless pipe dreams. For vexillary permutations, we then construct a new model that we call the semidual version model. We use our semidual model and the five-vertex model of Motegi and Sakai to given a new proof that double Grothendieck polynomials for vexillary permutations are equal to flagged factorial Grothendieck polynomials. Taking the stable limit of double Grothendieck polynomials, we obtain a new proof that the stable limit is a factorial Grothendieck polynomial as defined by McNamara. The states of our semidual model naturally correspond to families of nonintersecting lattice paths, where we can then use the Lindstr\"om-Gessel-Viennot lemma to give a determinant formula for double Schubert polynomials corresponding to vexillary permutations.