Saturation of Newton polytopes of type A and D cluster variables

TL;DR

This paper investigates the geometric properties of Newton polytopes associated with cluster variables in types A and D cluster algebras, revealing saturation conditions and characterizing when these polytopes are empty.

Contribution

It proves that all cluster variable Newton polytopes are saturated under certain conditions and characterizes when they are empty, advancing understanding of their geometric structure.

Findings

Newton polytopes are saturated with principal coefficients or boundary frozen variables

Characterization of when Newton polytopes are empty

Insights into the geometric structure of cluster variables in types A and D

Abstract

We study Newton polytopes for cluster variables in cluster algebras $\mathcal{A}(\Sigma)$ of types A and D. A famous property of cluster algebras is the Laurent phenomenon: each cluster variable can be written as a Laurent polynomial in the cluster variables of the initial seed $\Sigma$. The cluster variable Newton polytopes are the Newton polytopes of these Laurent polynomials. We show that if $\Sigma$ has principal coefficients or boundary frozen variables, then all cluster variable Newton polytopes are saturated. We also characterize when these Newton polytopes are \emph{empty}; that is, when they have no non-vertex lattice points.