Polyhedral models for generalized associahedra via Coxeter elements

TL;DR

This paper generalizes the construction of generalized associahedra, convex polytopes linked to cluster algebras, to any orientation of Dynkin diagrams, and shows consistency with existing Cambrian fan models.

Contribution

It extends the explicit realization of generalized associahedra to all orientations and aligns this with Cambrian fan constructions.

Findings

Generalized associahedra constructed for any orientation.

Confirmed agreement with Cambrian fan models.

Extended parametrization of cluster variables via g-vectors.

Abstract

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 \emph{generalized associahedron}. They provided an explicit realization of this polytope 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 construction 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.