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 −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 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.