On the Properties of the Exchange Graph of a Cluster Algebra

We prove a conjecture about the vertices and edges of the exchange graph of a cluster algebra A in two cases: when A is of geometric type and when A is arbitrary and its exchange matrix is nondegenerate. In the second case we also prove that the exchange graph does not depend on the coefficients of A. Both conjectures were formulated recently by Fomin and Zelevinsky. 1. Main definitions and results To state our results, recall the definition of a cluster algebra; for details see [FZ1, FZ4]. Let P be a semifield, that is, a free multiplicative abelian group of a finite rank m with generators g1, . . . , gm endowed with an additional operation ⊕, which is commutative, associative and distributive with respect to the multiplication. As an ambient field we take the field F of rational functions in n independent variables with coefficients in the field of fractions of the integer group ring ZP . A square matrix B is called skew-symmetrizable if DB is skew-symmetric for a non-degenerate non-negative diagonal matrix D. A seed is a triple Σ = (x,y, B), where x = (x1, . . . , xn) is a transcendence basis of F over the field of fractions of ZP , y = (y1, . . . , yn) is an n-tuple of elements of P and B is a skew-symmetrizable integer n × n matrix. The components of the seed are called the cluster , the coefficient tuple and the exchange matrix , respectively; the entries of x are called cluster variables. A seed mutation in direction k ∈ [1, n] takes Σ to an adjacent seed Σ = (x,y, B) whose components are defined as follows. The adjacent cluster x is given by x = (x \ {xk})∪{x ′ k}, where the new cluster variable x ′ k is defined by the exchange relation (1) xkx ′ k = yj yj ⊕ 1 ∏ bki>0 xki i + 1 yj ⊕ 1 ∏