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Realizations of the associahedron and cyclohedron. (English) Zbl 1125.52011
The associahedron first arose purely as the graph corresponding to the bistellar operations on triangulations of a convex polygon which only use the vertices of the polygon. Later, it was shown how to realize this graph as the edge-graph of a suitable simple convex polytope. Indeed, this polytope can be taken to be the secondary polytope (a special case of a fibre polytope) of the projection of an appropriate simplex on the polygon. In a similar way, the cyclohedron corresponds to centrally symmetric triangulations of a centrally symmetric polygon. There is also a connexion with the permutahedra arising from the Coxeter groups of types $$A$$ and $$B$$, respectively.
In this paper, the authors discuss realizations of these polytopes with integer coordinates for their vertices, comparing them with these permutahedra; these coordinates are obtained through an algorithm which uses the corresponding oriented Coxeter graphs of types $$A$$ and $$B$$ as the only input data.

##### MSC:
 52B12 Special polytopes (linear programming, centrally symmetric, etc.) 52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
##### Keywords:
associahedron; permutahedron; cyclohedron; Coxeter graph
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