2014/04/30 by Changjian Fu · 1 citation
Mathematics · #math.RA #math.RT
paper · pdf · doi:10.1016/j.jalgebra.2016.10.031
published as Journal of Algebra 473(2017) 194-220 · 27pages, references added, corrected some typos
arxiv created 2015/10/29 · arxiv updated 2018/09/11
Inspired by the tropical duality in cluster algebras, we introduce c-vectors for finite-dimensional algebras via τ-tilting theory. Let A be a finite-dimensional algebra over a 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 of this kind of c-vectors. We then study the positive c-vectors for certain classes of finite-dimensional algebras. More precisely, we establish the equalities between the set of positive c-vectors and the set of dimension vectors of exceptional modules for quasitilted algebras and representation-directed algebras respectively. This generalizes the equalitites of c-vectors for acyclic cluster algebras obtained by 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.