Many non-equivalent realizations of the associahedron

TL;DR

This paper classifies various realizations of the associahedron constructed by different methods, showing that only the Chapoton-Fomin-Zelevinsky realization can be obtained by both main methods, and introduces a new Santos construction.

Contribution

It provides a classification of associahedra realizations from two known methods and introduces the Santos construction, relating it to cluster complexes and denominator fans.

Findings

Only the Chapoton-Fomin-Zelevinsky associahedron is obtainable by both methods.

Two Hohlweg-Lange associahedra are linearly equivalent iff they are isometric.

The Santos construction is related to c-cluster complexes and denominator fans.

Abstract

Hohlweg and Lange (2007) and Santos (2004, unpublished) have found two different ways of constructing exponential families of realizations of the n-dimensional associahedron with normal vectors in {0,1,-1}^n, generalizing the constructions of Loday (2004) and Chapoton-Fomin-Zelevinsky (2002). We classify the associahedra obtained by these constructions modulo linear equivalence of their normal fans and show, in particular, that the only realization that can be obtained with both methods is the Chapoton-Fomin-Zelevinsky (2002) associahedron. For the Hohlweg-Lange associahedra our classification is a priori coarser than the classification up to isometry of normal fans, by Bergeron-Hohlweg-Lange-Thomas (2009). However, both yield the same classes. As a consequence, we get that two Hohlweg-Lange associahedra have linearly equivalent normal fans if and only if they are isometric. The Santos construction, which produces an even larger family of associahedra, appears here in print for the first time. Apart of describing it in detail we relate it with the c-cluster complexes and the denominator fans in cluster algebras of type A. A third classical construction of the associahedron, as the secondary polytope of a convex n-gon (Gelfand-Kapranov-Zelevinsky, 1990), is shown to never produce a normal fan linearly equivalent to any of the other two constructions.