c-vectors via τ-tilting theory

Inspired by the tropical dualities in cluster algebras, we introduce c-vectors for finite-dimensional algebras via τ-tilting theory. Let A be a finite-dimensional algebra over an algebraically closed field k. Each c-vector of A can be realized as the (negative) dimension vector of certain indecomposable A-module and hence we establish the sign-coherence property for this kind of c-vectors. We then study the positive c-vectors for certain classes of finite-dimensional algebras including quasitilted algebras and cluster-tilted algebras. In particular, we recover the equalities of c-vectors for acyclic cluster algebras and skew-symmetric cluster algebras of finite type respectively obtained by Nájera Chávez. To this end, a short proof for the sign-coherence of c-vectors for skew-symmetric cluster algebras has been given in the appendix.