The brick polytope of a sorting network

The associahedron is a polytope whose graph is the graph of flips on triangulations of a convex polygon. Pseudotriangulations and multitriangulations generalize triangulations in two different ways, which have been unified by Pilaud and Pocchiola in their study of flip graphs on pseudoline arrangements with contacts supported by a given sorting network. In this paper, we construct the brick polytope of a sorting network, obtained as the convex hull of the brick vectors associated to each pseudoline arrangement supported by the network. We combinatorially characterize the vertices of this polytope, describe its faces, and decompose it as a Minkowski sum of matroid polytopes. Our brick polytopes include Hohlweg and Lange's many realizations of the associahedron, which arise as brick polytopes for certain well-chosen sorting networks. We furthermore discuss the brick polytopes of sorting networks supporting pseudoline arrangements which correspond to multitriangulations of convex polygons: our polytopes only realize subgraphs of the flip graphs on multitriangulations and they cannot appear as projections of a hypothetical multiassociahedron.Comment: 36 pages, 25 figures; Version 2 refers to the recent generalization of our results to spherical subword complexes on finite Coxeter groups (http://arxiv.org/abs/1111.3349

Polytopal realizations of finite type g\mathbf{g}-vector fans

This paper shows the polytopality of any finite type $\mathbf{g}$-vector fan, acyclic or not. In fact, for any finite Dynkin type $\Gamma$, we construct a universal associahedron $\mathsf{Asso}_{\mathrm{un}}(\Gamma)$ with the property that any $\mathbf{g}$-vector fan of type $\Gamma$ is the normal fan of a suitable projection of $\mathsf{Asso}_{\mathrm{un}}(\Gamma)$.Comment: 27 pages, 9 figures; Version 2: Minor changes in the introductio