Combinatorial Theory

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 empty; that is, when they have no non-vertex lattice points.

Mathematics Subject Classifications: 13F60, 52B20

Keywords: Cluster algebras, Newton polytopes, snake graphs