Combinatorics of $F$-polynomials

Abstract

We introduce the stabilization functors to study the combinatorial aspect of the F -polynomial of a representation of any finite-dimensional basic algebra. We characterize the vertices of their Newton polytopes. We give an explicit formula for the F -polynomial restricting to any face of its Newton polytope. For acyclic quivers, we give a complete description of all facets of the Newton polytope when the representation is general. We also prove that the support of F -polynomial is saturated for any rigid representation. We provide many examples and counterexamples, and pose several conjectures.

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

“…Then as a conclusion, in Corollary 4.5, we provide a positive answer to Conjecture 4.4 posed in [6] by Fei. On the other hand, when A is in particular skew-symmetrizable, we can calculate the cluster algebra associated to each face S by the following result: ♠ (Theorem 4.8) In Theorem 3.9 (v), if A is a skew-symmetrizable cluster algebra with principal coefficients, and denote by B ′ the initial exchange matrix of the cluster algebra A ′ , then B ′ = W ⊤ BW , where W = (τ (e 1 ) ⊤ , • • • , τ (e r ) ⊤ ), W = (τ (e 1 ) ⊤ , • • • , τ (e r ) ⊤ ) are n × r integer matrices, w ji e j , with s = 0 being the label of the edge in S parallel to τ (e i ) while τ (e i ) = r j=1 d j w ji e j when the label is 0.…”

“…Newton polytopes of F -polynomials and recurrence formula on cluster variables. We have known in [6] that the Newton polytope of an F -polynomial is defined associated to representations of a finite-dimensional basic algebra, as well as some interesting combinatorial properties of these Newton polytopes. But those cluster algebras, whose categorification have not been found so far, are not suitable for the theory in [6].…”

“…We have known in [6] that the Newton polytope of an F -polynomial is defined associated to representations of a finite-dimensional basic algebra, as well as some interesting combinatorial properties of these Newton polytopes. But those cluster algebras, whose categorification have not been found so far, are not suitable for the theory in [6]. In this sense, it is necessary for us to establish the theory of these Newton polytopes for general TSSS cluster algebras.…”

“…First we introduce some definitions from [6], then we will obtain some results analogous to those in [6], but in the context of totally sign-skew-symmetric cluster algebras.…”

“…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.…”