Cluster algebras and symmetric matrices

Abstract:

In the structural theory of cluster algebras, a crucial role is played by a family of integer vectors, called $\mathbf {c}$-vectors, which parametrize the coefficients. It has recently been shown that each $\mathbf {c}$-vector with respect to an acyclic initial seed is a real root of the corresponding root system. In this paper, we obtain an interpretation of this result in terms of symmetric matrices. We show that for skew-symmetric cluster algebras, the $\mathbf {c}$-vectors associated with any seed defines a quasi-Cartan companion for the corresponding exchange matrix (i.e. they form a companion basis), and we establish some basic combinatorial properties. In particular, we show that these vectors define an admissible cut of edges in the associated quivers.