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arXiv · 2012.07500

Saturation of Newton polytopes of type A and D cluster variables

Amal Mattoo·Melissa Sherman-Bennett
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Abstract

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 \emph{empty}; that is, when they have no non-vertex lattice points.

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Amal Mattoo, Melissa Sherman-Bennett. 2020-12-14. Saturation of Newton polytopes of type A and D cluster variables. https://arxiv.org/abs/2012.07500

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