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arXiv · 1910.14372

Newton polytopes of rank 3 cluster variables

Kyungyong Lee·Li Li·Ralf Schiffler
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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.

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Kyungyong Lee, Li Li, Ralf Schiffler. 2019-10-31. Newton polytopes of rank 3 cluster variables. https://arxiv.org/abs/1910.14372

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