F -Matrices of Cluster Algebras from Triangulated Surfaces
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Annals of Combinatorics
F -Matrices of Cluster Algebras from Triangulated Surfaces Yasuaki Gyoda and Toshiya Yurikusa Abstract. For a given marked surface (S, M ) and a fixed tagged triangulation T of (S, M ), we show that each tagged triangulation T of (S, M ) is uniquely determined by the intersection numbers of tagged arcs of T and tagged arcs of T . As a consequence, each cluster in the cluster algebra A(T ) is uniquely determined by its F -matrix which is a new numerical invariant of the cluster introduced by Fujiwara and Gyoda. Keywords. Marked surface, Tagged triangulation, Intersection number, Cluster algebra, F -matrix.
1. Introduction Cluster algebras are commutative subrings of rational function fields. They were introduced in [9] to study total positivity of semisimple Lie groups and canonical bases of quantum groups. Nowadays, it is found that cluster algebras appear in various subjects in mathematics, for example, representation theory of quivers, Poisson geometry, integrable systems, and so on. One of important classes of cluster algebras is given from marked surfaces that were developed in [5–8,14]. For a marked surface (S, M ) and the associated cluster algebra, its cluster complex is identified with a connected component of the tagged arc complex of (S, M ) [7]. In this way, cluster variables correspond to tagged arcs, and clusters correspond to tagged triangulations. Many properties of the cluster algebra can be shown using this correspondence (see, e.g. [3,7,8,17–20]). Qiu and Zhou [22] introduced an intersection number of two tagged arcs to study cluster categories. The aim of this paper was to study a new numerical invariant of cluster variables and clusters, called f -vectors and F -matrices, respectively, introduced in [12,13] for the cluster algebra associated with (S, M ). To do so, we use intersection numbers of tagged arcs since it was proved by [25] that intersection vectors coincide with f -vectors in the associated cluster algebra. We fix 0123456789().: V,-vol
Y. Gyoda, T. Yurikusa
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Figure 1. Pairs (δ, ) of conjugate arcs a tagged triangulation T of (S, M ). For a tagged arc δ of (S, M ), we consider a vector, called its intersection vector, whose entries are intersection numbers of δ and tagged arcs of T . We show that a tagged triangulation T of (S, M ) is uniquely determined by the intersection vectors of tagged arcs of T (Theorem 1.1). It induces our main result: in this case, clusters are uniquely determined by their F -matrices (Corollary 4.9). This paper is organized as follows: In the rest of this section, we give the results of this paper. In Sect. 2, we prove our results Theorems 1.1 and 1.3 below. For that reason, we introduce modifications of tagged arcs. It plays a key role in our proofs that they are uniquely determined by their intersection vectors (Theorem 2.5). In Sect. 3, we study a more detailed result of Theorem 1.1. In Sect. 4, we recall the notions of f -vectors and F -matrices. Using the correspondence between f -vectors and intersection vectors gi
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