Diagrammatic description of \(c\)-vectors and \(d\)-vectors of cluster algebras of finite type (Q405060)

Summary: 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 \(A_n\) and \(D_n\), then we apply the folding method to \(D_{n+1}\) and \(A_{2n-1}\) to obtain types \(B_n\) and \(C_n\). 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.