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Coxeter-associahedra

Part of: Polytopes and polyhedra

Published online by Cambridge University Press:  26 February 2010

Victor Reiner  and
Günter M. Ziegler
Victor Reiner
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, U.S.A.
Günter M. Ziegler
Affiliation:
Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB), Heilbronner Str. 10, D-10711 Berlin, Germany
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Abstract

Recently M. M. Kapranov [Kap] defined a poset KPAn−1, called the permuto-associahedron, which is a hybrid between the face poset of the permutohedron and the associahedron. Its faces are the partially parenthesized, ordered, partitions of the set {1, 2, …, n}, with a natural partial order.

Kapranov showed that KPAn−1, is the face poset of a regular CW-ball, and explored its connection with a category-theoretic result of MacLane, Drinfeld's work on the Knizhnik-Zamolodchikov equations, and a certain moduli space of curves. He also asked the question of whether this CW-ball can be realized as a convex polytope.

We show that indeed, the permuto-associahedron corresponds to the type An−1, in a family of convex polytopes KPW associated to the classical Coxeter groups, W = An−1, Bn, Dn. The embedding of these polytopes relies on the secondary polytope construction of the associahedron due to Gel'fand, Kapranov, and Zelevinsky. Our proofs yield integral coordinates, with all vertices on a sphere, and include a complete description of the facet-defining inequalities.

Also we show that for each W, the dual polytope KPW* is a refinement (as a CW-complex) of the Coxeter complex associated to W, and a coarsening of the barycentric subdivision of the Coxeter complex. In the case W = An−1, this gives a combinatorial proof of Kapranov's original sphericity result.

MSC classification

Secondary: 52B05: Combinatorial properties (number of faces, shortest paths, etc.)
Type
Research Article
Information
Mathematika , Volume 41 , Issue 2 , December 1994 , pp. 364 - 393
Copyright
Copyright © University College London 1994

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