Many non-equivalent realizations of the associahedron

Details Many non-equivalent realizations of the associahedron Many non-equivalent realizations of the associahedron

2011-09-26

arxiv.org/abs/1109.5544v1

Mathematics

Metric Geometry

28 pages, 15 figures

Scientific paper

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.

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