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∈ℤ 3 such that the Laurent monomial x p appears with nonzero coefficient in the Laurent expansion of z in the cluster x. We give an explicit construction of the Newton polytope in terms of the exchange matrix and the denominator vector of the cluster variable.
Highlights
Cluster algebras were discovered by Fomin and Zelevinsky in 2001
It has been shown that they are related to diverse areas of mathematics such as algebraic geometry, total positivity, quiver representations, string theory, statistical physics models, non-commutative geometry, Teichmüller theory, tropical geometry, KP solitons, discrete integrable systems, quantum mechanics, Lie theory, algebraic combinatorics, WKB analysis, knot theory, number theory, symplectic geometry, and Poisson geometry
Explicit combinatorial formulas that are manifestly positive are known for cluster variables in cluster algebras from surfaces [16] and for cluster algebras of rank 2 [14]
Summary
Cluster algebras were discovered by Fomin and Zelevinsky in 2001. A cluster algebra is equipped with a set of distinguished generators called cluster variables. These generators are very far from being fully understood. Explicit combinatorial formulas that are manifestly positive are known for cluster variables in cluster algebras from surfaces [16] and for cluster algebras of rank 2 [14]. For skew-symmetric cluster algebras, there is the cluster character formula for the cluster variables [18] as well as the F -polynomial formula [6] and for skew-symmetrizable cluster algebras there is the scattering diagram approach [10], but none of these provide computable formulas.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
