# Many non-equivalent realizations of the associahedron

## Cesar Ceballos, Francisco Santos, Günter M. Ziegler – 2011

Focus Area 1: High-complexity Geometry We show that three systematic construction methods for the $n$-dimensional associahedron, - as the secondary polytope of a convex $(n+3)$-gon (by Gelfand-Kapranov-Zelevinsky), - via cluster complexes of the root system $A_n$ (by Chapoton-Fomin-Zelevinsky), and - as Minkowski sums of simplices (by Postnikov) produce substantially different realizations, independent of the choice of the parameters for the constructions. The cluster complex and the Minkowski sum realizations were generalized by Hohlweg-Lange to produce exponentionally many distinct realizations, all of them with normal vectors in $\{0,\pm1\}^n$. We present another, even larger, exponential family, generalizing the cluster complex construction -- and verify that this family is again disjoint from the previous ones, with one single exception: The Chapoton-Fomin-Zelevinsky associahedron appears in both exponential families.