Lattice Polytopes from Schur and Symmetric Grothendieck Polynomials

Given a family of lattice polytopes, two common questions in Ehrhart Theory are determining when a polytope has the integer decomposition property and determining when a polytope is reflexive. While these properties are of independent interest, the confluence of these properties is a source of active investigation due to conjectures regarding the unimodality of the $h^\ast$-polynomial. In this paper, we consider the Newton polytopes arising from two families of polynomials in algebraic combinatorics: Schur polynomials and inflated symmetric Grothendieck polynomials. In both cases, we prove that these polytopes have the integer decomposition property by using the fact that both families of polynomials have saturated Newton polytope. Furthermore, in both cases, we provide a complete characterization of when these polytopes are reflexive. We conclude with some explicit formulas and unimodality implications of the $h^\ast$-vector in the case of Schur polynomials.

Double Grothendieck Polynomials and Colored Lattice Models

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 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 give 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 Lindström–Gessel–Viennot lemma to give a determinant formula for double Schubert polynomials corresponding to vexillary permutations.