Cluster algebras and symmetrizable matrices

Abstract

In the structure theory of cluster algebras, principal coefficients are parametrized by a family of integer vectors, called $\mathbf {c}$-vectors. Each $\mathbf {c}$-vector with respect to an acyclic initial seed is a real root of the corresponding root system, and the $\mathbf {c}$-vectors associated with any seed defines a symmetrizable quasi-Cartan companion for the corresponding exchange matrix. We establish basic combinatorial properties of these companions. In particular, we show that $\mathbf {c}$-vectors define an admissible cut of edges in the associated diagrams.