COXETER-ASSOCIAHEDRA

Recently M. M. Kapranov [Kap] defined a poset KPA(n - I), called the p ermuto-associahedron, which is a hybrid between the face poset of the permutohedron and the associahedron. Its faces are the partially paren thesized, ordered, partitions of the set {1, 2,..., n}, with a natural partial order. Kapranov showed that KPA(n - 1) is the face poset of a regular CW-ball, and explored its connection with a category-theoreti c result of MacLane, Drinfeld's work on the Knizhnik-Zamolodchikov equ ations, and a certain moduli space of curves. He also asked the questi on of whether this CW-ball can be realized as a convex polytope. We sh ow that indeed, the permuto-associahedron corresponds to the type A(n - 1) in a family of convex polytopes KPW associated to the classical C oxeter groups, W = A(n - 1), B-n, D-n. The embedding of these polytope s relies on the secondary polytope construction of the associahedron d ue to Gel'fand, Kapranov, and Zelevinsky. Our proofs yield integral co ordinates, with all vertices on a sphere, and include a complete descr iption 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 Cox eter complex associated to W, and a coarsening of the barycentric subd ivision of the Coxeter complex. In the case W = A(n - 1), this gives a combinatorial proof of Kapranov's original sphericity result.