Newton polytopes of rank 3 cluster variables

2019/10/31 by Lee, Kyungyong, Li, Li, Schiffler, Ralf
#Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Primary 13F60 #Secondary 52B20

paper · doi:10.48550/arxiv.1910.14372

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 p∈ℤ³ such that the Laurent monomial \bf x^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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