Saturation of Newton polytopes of type A and D cluster variables

Abstract

Cluster algebras are rings with distinguished generators called cluster variables, grouped into clusters. Each cluster variable can be written as a Laurent polynomial in any cluster. In this paper, we study the Newton polytopes of these Laurent polynomials. We focus on cluster algebras of types A and D, with frozen variables corresponding to the boundary segments of a polygon and punctured polygon, respectively. For these cluster algebras, we show the cluster variable Newton polytopes are saturated. For type A, we additionally show that the cluster variable Newton polytopes have no non-vertex lattice points. Our main tool is the snake graph expansion formula of Musiker-Schiffler-Williams for cluster algebras from surfaces.