Polyhedral models for generalized associahedra via Coxeter elements

Motivated by the theory of cluster algebras, F. Chapoton, S. Fomin, and A. Zelevinsky associated to each finite type root system a simple convex polytope, called generalized associahedron. They provided an explicit realization of this poly- tope associated with a bipartite orientation of the corresponding Dynkin diagram. In the first part of this paper, using the parametrization of cluster variables by their g-vectors explicitly computed by S.-W. Yang and A. Zelevinsky, we generalize the original construction to any orientation. In the second part we show that our construc- tion agrees with the one given by C. Hohlweg, C. Lange, and H. Thomas in the setup of Cambrian fans developed by N. Reading and D. Speyer.