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.

Convex-normal (Pairs of) Polytopes

In 2012, Gubeladze (Adv. Math. 2012) introduced the notion ofk-convex-normal polytopes to show that integral polytopes all of whose edges are longer than 4d(d+ 1) have the integer decomposition property. In the first part of this paper we show that for lattice polytopes there is no diòerence betweenk- and (k+ 1)-convex-normality (fork≥ 3) and improve the bound to 2d(d+ 1). In the second part we extend the definition to pairs of polytopes. Given two rational polytopesPandQ, where the normal fan ofPis a reûnement of the normal fan ofQ, if every edge ePofPis at least d times as long as the corresponding face (edge or vertex) eQofQ, then.

Integer Decomposition Property of Dilated Polytopes

An integral convex polytope $\mathcal{P} \subset \mathbb{R}^N$ possesses the integer decomposition property if, for any integer $k > 0$ and for any $\alpha \in k \mathcal{P} \cap \mathbb{Z}^{N}$, there exist $\alpha_{1}, \ldots, \alpha_k \in \mathcal{P} \cap \mathbb{Z}^{N}$ such that $\alpha = \alpha_{1} + \cdots + \alpha_k$. A fundamental question is to determine the integers $k > 0$ for which the dilated polytope $k\mathcal{P}$ possesses the integer decomposition property. In the present paper, combinatorial invariants related to the integer decomposition property of dilated polytopes will be proposed and studied.

Edge ideals of clique clutters of comparability graphs and the normality of monomial ideals

The normality of a monomial ideal is expressed in terms of lattice points of blocking polyhedra and the integer decomposition property. For edge ideals of clutters this property characterizes normality. Let $G$ be the comparability graph of a finite poset. If $\mathrm{cl}(G)$ is the clutter of maximal cliques of $G$, we prove that $\mathrm{cl}(G)$ satisfies the max-flow min-cut property and that its edge ideal is normally torsion free. Then we prove that edge ideals of complete admissible uniform clutters are normally torsion free.