Abstract:

The sign-coherence property of $c$-vectors plays an important role in the structure theory of cluster algebras. However, all known proofs depend on either the invariant theory of geometry or the representation theory. In this paper we give an elementary proof for the sign-coherence property of $c$-vectors for cluster algebras of rank $2$. As an application, we prove that a cluster is determined by its associated subset of positive $c$-vectors.

Key words: Cluster algebra, $c$-vector, Sign-coherence