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

Abstract

In this paper, we study the Newton polytopes of F -polynomials in totally sign-skewsymmetric cluster algebras and generalize them to a larger set consisting of polytopes N h associated to vectors h ∈ Z n as well as P consisting of polytope functions ρ h corresponding to N h .The main contribution contains that (i) obtaining a recurrence construction of the Laurent expression of a cluster variable in a cluster from its g-vector; (ii) proving the subset P of P is a strongly positive basis of U (A) consisting of certain universally indecomposable Laurent polynomials, which is called as the polytope basis; (iii) constructing some explicit maps among corresponding F -polynomials, g-vectors and d-vectors to characterize their relationship.As an application of (i), we give an affirmation to the positivity conjecture of cluster variables in a totally sign-skew-symmetric cluster algebra, which in particular provides a new method different from that given in [12] to present the positivity of cluster variables in the skew-symmetrizable case.As another application, a conjecture on Newton polytopes posed by Fei is answered affirmatively.For (ii), we know that in rank 2 case, P coincides with the greedy basis introduced by Lee, Li and Zelevinsky. Hence, we can regard P as a natural generalization of the greedy basis in general rank.As an application of (iii), the positivity of denominator vectors associated to non-initial cluster variables, which was first come up as a conjecture in [10], is proved in a totally sign-skew-symmetric cluster algebra.

Knot theory and cluster algebras

“…The Laurent phenomenon was proved in [18] for sign-skew symmetric algebras. Positivity and sign-coherence were shown in [23] in the skew-symmetrizable and in [28] in the sign-skew symmetric case. Theorems 2.5 and 2.6 hold in the skew-symmetrizable case by [20] and a variation of Theorem 2.8 holds in this case as well [33].…”

“…Newton polytopes. This application uses results from [28]. The Newton polytope of a multivariate polynomial c (a1,...,a N ) y a1 1 • • • y a N N is the convex hull of the of the lattice points (a 1 , .…”

“…(1) The Newton polytope is saturated, that is, the lattice points in the Newton polytope are precisely the points e for which the coefficient c e is nonzero. Moreover, the coefficient c e is equal to 1 if and only if the point e is a vertex of the Newton polytope [28,Theorem 4.4]. (2) The k-th component of the denominator vector of the Laurent expansion of the cluster variable u is equal to the maximum length of an edge of the Newton polytope that is parallel to the k-th standard basis vector of R n [28, Theorem 5.4].…”

The valuation pairing on an upper cluster algebra

“…It is known that F -polynomials enjoy many nice properties, for example, they have positive coefficients and constant term 1. We refer the readers to [LS15, DWZ10, GHKK18] for these results and to [GY20,Gyo21,FG19a,Fei19a,Fei19b,LP22] for some recent work on F -polynomials.…”

Saturation of Newton polytopes of type A and D cluster variables

“…Cluster variable Newton polytopes for principal coefficient cluster algebras are related to those for cluster algebras with arbitrary frozen variables by a linear map given by the columns of the extended exchange matrix. However, this map is not usually unimodular, so may not preserve saturation.4 After this manuscript appeared, Fei's conjecture was proven in full generality in[LP22].…”