IDP for 2-Partition Maximal Symmetric Polytopes

TL;DR

The work tackles whether every symmetric lattice polytope has the integer decomposition property (IDP) by developing a framework that connects saturated Newton polytopes (SNP) of symmetric polynomials to IDP of the corresponding Newton polytopes. It introduces the key construction $ts$—a sum of Schur polynomials indexed by $t$-fold sums of maximal partitions—and shows that, when $s$ has SNP, $ts$ has SNP for all $t$, implying $t\mathcal{P}=\operatorname{Newt}(ts)$ and hence IDP for all $t$. The authors specialize to 2-partition maximal polytopes in $\mathbb{R}^3$, using convex-hull and tableau-decomposition techniques to prove SNP for $ts$ and therefore IDP for the polytopes in this class. They also discuss conjectures for extending SNP-based IDP to more general symmetric polytopes and connect the framework to inflated symmetric Grothendieck polynomials and Schur-positivity. Overall, the paper provides a concrete methodology to certify IDP for symmetric polytopes via SNP-driven Newton-polytope analysis, with potential for broader applicability in combinatorial and geometric contexts.

Abstract

We provide a framework for which one can approach showing the integer decomposition property for symmetric polytopes. We utilize this framework to prove a special case which we refer to as $2$-partition maximal polytopes in the case where it lies in a hyperplane of $\mathbb{R}^3$. Our method involves proving a special collection of polynomials have saturated Newton polytope.

Figures (3)

• Figure 1:
• Figure 2: The first step of the decomposition process for a semistandard tableau of shape $(8,7,2)$.
• Figure 3: Steps two and three of the decomposition process of a tableau of shape $(8,7,2)$.

Theorems & Definitions (21)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Example 2.4
• Lemma 3.1
• Definition 3.2
• Remark 3.3
• Conjecture 3.4
• Theorem 3.5