Polyhedral models for generalized associahedra via Coxeter elements

Details Polyhedral models for generalized associahedra via Coxeter elements Polyhedral models for generalized associahedra via Coxeter elements

2011-11-07

arxiv.org/abs/1111.1657v2

Mathematics

Combinatorics

31 pages, 2 figures. Changelog: 20111106: initial version 20120403: fixed errors in figures

Scientific paper

Motivated by the theory of cluster algebras, S. Fomin and A. Zelevinsky have associated to each finite type root system a simple convex polytope called generalized associahedron. We give a new family of geometric realizations of these polytopes, associated with arbitrary orientations of the Dynkin diagram. Our construction uses a parametrization of cluster variables by their $g$-vectors explicitly computed by S.-W. Yang and A. Zelevinsky. We also show that our construction agrees with the one given by C. Hohlweg, C. Lange, H. Thomas in the setup of Cambrian fans developed by N. Reading and D. Speyer.

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