Newton polytopes of rank 3 cluster variables

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.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.

Recurrence formula, positivity and polytope basis in cluster algebras via Newton polytopes

“…Remark 3.10. The analogue conclusions of (ii) and (iv) were proved in [18] and [17] for Newton polytopes of cluster variables in rank 2 and rank 3 cases respectively. Note that this is not a coincidence.…”

“…According to the inductive construction and Corollary 3.15 proved below for n − 1 case, such α ′ and w ′ uniquely exist. Similar to (17), we have…”

“…When A is a cluster algebra without coefficients of rank 2, we know in this case A = U(A) since A is acyclic. A Z-basis for U(A) is defined in [17] called the greedy basis {x…”

“…Such i, r exist because due to our assumption, N πj (h) | Aj has larger dimension than N πr(h) | Ar if N πr(h) | Ar lies in the hyperplane z j = 0. Then for the cluster algebra A ′ = A r , we have a decomposition (17) ρ πr(h) x i = ρ πr(h)+ei + j Y wj ρ αj for some w j ∈ N n−1 and α j ∈ Z n−1 by ( 13), ( 14) and the inductive construction of ρ πr(h) . Let U 0 h = j {N νj [γ r;0 (w j )] | ν j = γ r;hr+γr;0(wj )b ⊤ r (α j )} {N h+ei }, where h r is the r-th element of h and b ⊤ r is the r-th column of B ⊤ .…”

“…In [6], Jiarui Fei defined the Newton polytope of an F -polynomial associated to representations of a finite-dimensional basic algebra, as well as showed some interesting combinatorial properties of such Newton polytopes. On the other hand, the authors of [18] and [17] focused on Newton polytopes of cluster variables in cluster algebras of rank 2 and rank 3 respectively. By definitions, the Newton polytope of a cluster variable can be obtained from that of the related F -polynomial by a transformation induced by its exchange matrix B since ŷj;t = m i=1 x bji i;t in the case of geometric type.…”