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.
Highlights
Of central interest in algebraic combinatorics are polynomials f ∈ C[x1, x2, . . . , xm], which commonly appear as generating functions that encode some combinatorial information
We show that all Newton polytopes that arise from Schur polynomials have integer decomposition property (IDP), and characterize which of these are reflexive
We show that all Newton polytopes arising from inflated symmetric Grothendieck polynomials have IDP, and characterize the very few that are reflexive
Summary
Of central interest in algebraic combinatorics are polynomials f ∈ C[x1, x2, . . . , xm], which commonly appear as generating functions that encode some combinatorial information. A polynomial has saturated Newton polytope if every lattice point appearing in the Newton polytope corresponds to the exponent vector of a monomial in f with nonzero coefficient [19]. We study Newton polytopes arising from Schur polynomials and a generalization of symmetric Grothendieck polynomials, which we call inflated symmetric Grothendieck polynomials. We denote these polytopes by Newt(sλ) and Newt(Gh,λ), respectively. We show that all Newton polytopes that arise from Schur polynomials have IDP, and characterize which of these are reflexive. We show that all Newton polytopes arising from inflated symmetric Grothendieck polynomials have IDP, and characterize the very few that are reflexive
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