N ov 2 00 6 A cluster expansion formula ( A n case )

We consider the Ptolemy cluster algebras, which are cluster algebras of finite type A (with non-trivial coefficients) that have been described by Fomin and Zelevinsky using triangulations of a regular polygon. Given any seed Σ in a Ptolemy cluster algebra, we present a formula for the expansion of an arbitrary cluster variable in terms of the cluster variables of the seed Σ. Our formula is given in a combinatorial way, using paths on a triangulation of the polygon that corresponds to the seed Σ. 0 Introduction Ptolemy cluster algebras have been introduced by Fomin and Zelevinsky in [FZ2, section 12] as examples of cluster algebras of type A. The Ptolemy algebra of rank n is described using the triangulations of a regular polygon with n + 3 vertices. In this description, the seeds of the cluster algebra are in bijection with the triangulations of the polygon. The cluster of the seed corresponds to the diagonals, while the coefficients of the seed correspond to the boundary edges of the triangulation. The Laurent phenomenon [FZ1] states that, given an arbitrary seed Σ one can write any cluster variable of the cluster algebra as a Laurent-polynomial in the cluster variables and the coefficients of the seed Σ. The main result of this paper is an explicit formula for these Laurent polynomials, see Theorem 1.1. Each term of the Laurent polynomial is given by a path on the triangulation corresponding to the seed Σ. As an application, we prove the positivity conjecture of Fomin and Zelevinsky [FZ1] for Ptolemy algebras, see Corollary 1.6. The polygon model has also been used in [CCS] to construct the cluster category associated to the cluster algebra, compare [BMRRT]. In that context, the cluster algebra has trivial coefficients and our formula naturally applies to that situation, see Remark 1.4. Therefore, there is an interesting intersection with the work of Caldero and Chapoton [CC], who have obtained a formula for cluster expansions when the given seed is acyclic, meaning that each triangle in the triangulation has at least one side on the boundary of the polygon. Their formula, and its generalization by Caldero and Keller [CK], also applies to cluster algebras (with trivial coefficients) of other types, but, again, only in the case where the seed Σ is acyclic. Their description uses the representation theory of finite dimensional algebras and is very different from ours. 1 1 Cluster expansions in the Ptolemy algebra Throughout this paper, let n be a positive integer and let P be a regular polygon with n + 3 vertices. A diagonal in P is a line segment connecting two nonadjacent vertices of P and two diagonals are said to be crossing if they intersect in the interior of the polygon. A triangulation T is a maximal set of noncrossing diagonals together with all boundary edges. Any triangulation T has n diagonals and n + 3 boundary edges. Denote the boundary edges of P by Tn+1, . . . , T2n+3. 1.1 The Ptolemy cluster algebra We recall some facts about the Ptolemy cluster algebra of rank n from [FZ2, section 12.2]. The cluster variables xM of this algebra are in bijection with the diagonalsM of the polygon P , and the generators of its coefficient semifield are in bijection with the boundary edges Tn+1, . . . , T2n+3 of P . To be more precise, the coefficient semifield is the tropical semifield Trop(xn+1, xn+2, . . . , x2n+3), which is a free abelian group, written multiplicatively, with generators xn+1, . . . , x2n+3, and with auxiliary addition ⊕ given by