Lattice polytopes from Schur and symmetric Grothendieck polynomials

TL;DR

This paper investigates the geometric properties of Newton polytopes derived from Schur and symmetric Grothendieck polynomials, establishing their integer decomposition property and characterizing reflexivity, with implications for unimodality of associated polynomials.

Contribution

It proves that these Newton polytopes have the integer decomposition property and provides a complete characterization of their reflexivity, connecting algebraic combinatorics with polyhedral geometry.

Findings

Newton polytopes from Schur and Grothendieck polynomials have the integer decomposition property

Complete characterization of when these polytopes are reflexive

Explicit formulas and unimodality results for the $h^*$-vector in Schur polynomials

Abstract

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.