Many nonequivalent realizations of the associahedron
Abstract
Hohlweg and Lange (2007) and Santos (2004, unpublished) have found two different ways of constructing exponential families of realizations of the ndimensional associahedron with normal vectors in {0,1,1}^n, generalizing the constructions of Loday (2004) and ChapotonFominZelevinsky (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 ChapotonFominZelevinsky (2002) associahedron. For the HohlwegLange associahedra our classification is a priori coarser than the classification up to isometry of normal fans, by BergeronHohlwegLangeThomas (2009). However, both yield the same classes. As a consequence, we get that two HohlwegLange 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 ccluster 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 ngon (GelfandKapranovZelevinsky, 1990), is shown to never produce a normal fan linearly equivalent to any of the other two constructions.
 Publication:

arXiv eprints
 Pub Date:
 September 2011
 DOI:
 10.48550/arXiv.1109.5544
 arXiv:
 arXiv:1109.5544
 Bibcode:
 2011arXiv1109.5544C
 Keywords:

 Mathematics  Metric Geometry;
 Mathematics  Combinatorics;
 52B05;
 52B11
 EPrint:
 30 pages, 13 figures