Coxeter-Associahedra

Recently M. M. Kapranov [Kap] defined a poset KPA_n-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 KPA_n-1is 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 A_n-1 in a family of convex polytopes KPW associated to the classical Coxeter groups, W = A_n-1, B_n, D_n. 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 = A_n-1, this gives a combinatorial proof of Kapranov’s original sphericity result.