Newton polytopes of rank 3 cluster variables

TL;DR

This paper characterizes the Newton polytopes of rank 3 cluster variables in skew-symmetrizable cluster algebras, providing explicit constructions and a new proof regarding denominator vectors.

Contribution

It offers an explicit construction of Newton polytopes for rank 3 cluster variables and proves non-negativity of denominator vectors in arbitrary rank.

Findings

Newton polytope of a cluster variable is the convex hull of certain lattice points.

Explicit construction of Newton polytope from exchange matrix and denominator vector.

Non-negativity of denominator vectors for non-initial cluster variables.

Abstract

We characterize the cluster variables of skew-symmetrizable cluster algebras of rank 3 by their Newton polytopes. The Newton polytope of the cluster variable $z$ is the convex hull of the set of all $\mathbf{p}\in\mathbb{Z}^3$ such that the Laurent monomial ${\bf x}^{\mathbf{p}}$ appears with nonzero coefficient in the Laurent expansion of $z$ in the cluster ${\bf x}$. We give an explicit construction of the Newton polytope in terms of the exchange matrix and the denominator vector of the cluster variable. Along the way, we give a new proof of the fact that denominator vectors of non-initial cluster variables are non-negative in a cluster algebra of arbitrary rank.