Diagrammatic description of c-vectors and d-vectors of cluster algebras of finite type to the memory of andrei zelevinsky

Abstract

We provide an explicit Dynkin diagrammatic description of the c-vectors and the d-vectors (the denominator vectors) of any cluster algebra of finite type with principal coefficients and any initial exchange matrix. We use the surface realization of cluster algebras for types An and Dn, then we apply the folding method to Dn+1 and A2n-1 to obtain types Bn and Cn. Exceptional types are done by direct inspection with the help of a computer algebra software. We also propose a conjecture on the root property of c-vectors for a general cluster algebra.