ALGEBRAIC COMBINATORICS

Newton polytopes of rank 3 cluster variables
Algebraic Combinatorics, Volume 3 (2020) no. 6, pp. 1293-1330.

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 $\mathbf{p}\in {ℤ}^{3}$ such that the Laurent monomial ${\mathbf{x}}^{\mathbf{p}}$ appears with nonzero coefficient in the Laurent expansion of $z$ in the cluster $\mathbf{x}$. 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.

Revised:
Accepted:
Published online:
DOI: https://doi.org/10.5802/alco.141
Classification: 13F60,  52B20
Keywords: Newton polytopes, cluster variables
@article{ALCO_2020__3_6_1293_0,
author = {Lee, Kyungyong and Li, Li and Schiffler, Ralf},
title = {Newton polytopes of rank 3 cluster variables},
journal = {Algebraic Combinatorics},
pages = {1293--1330},
publisher = {MathOA foundation},
volume = {3},
number = {6},
year = {2020},
doi = {10.5802/alco.141},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.141/}
}
Lee, Kyungyong; Li, Li; Schiffler, Ralf. Newton polytopes of rank 3 cluster variables. Algebraic Combinatorics, Volume 3 (2020) no. 6, pp. 1293-1330. doi : 10.5802/alco.141. https://alco.centre-mersenne.org/articles/10.5802/alco.141/

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