Saturation of Newton polytopes of type A and D cluster variables

Abstract

We study Newton polytopes for cluster variables in cluster algebras A(Σ) 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 Σ. The cluster variable Newton polytopes are the Newton polytopes of these Laurent polynomials. We show that if Σ 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.