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Realizations of the associahedron and cyclohedron

Christophe Hohlweg·Carsten Lange
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Abstract

We describe many different realizations with integer coordinates for the associahedron (i.e. the Stasheff polytope) and for the cyclohedron (i.e. the Bott-Taubes polytope) and compare them to the permutahedron of type A_n and B_n respectively. The coordinates are obtained by an algorithm which uses an oriented Coxeter graph of type A_n or B_n respectively as only input and which specialises to a procedure presented by J.-L. Loday for a certain orientation of A_n. The described realizations have cambrian fans of type A and B as normal fans. This settles a conjecture of N. Reading for cambrian fans of these types.

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Christophe Hohlweg, Carsten Lange. 2005-10-27. Realizations of the associahedron and cyclohedron. https://arxiv.org/abs/math/0510614

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