The local hh-vector of the cluster subdivision of a simplex

The cluster complex $\Delta (\Phi)$ is an abstract simplicial complex, introduced by Fomin and Zelevinsky for a finite root system $\Phi$. The positive part of $\Delta (\Phi)$ naturally defines a simplicial subdivision of the simplex on the vertex set of simple roots of $\Phi$. The local $h$-vector of this subdivision, in the sense of Stanley, is computed and the corresponding $\gamma$-vector is shown to be nonnegative. Combinatorial interpretations to the entries of the local $h$-vector and the corresponding $\gamma$-vector are provided for the classical root systems, in terms of noncrossing partitions of types $A$ and $B$. An analogous result is given for the barycentric subdivision of a simplex.Comment: 21 pages, 4 figure

h-vectors of generalized associahedra and non-crossing partitions

A case-free proof is given that the entries of the $h$-vector of the cluster complex $\Delta (\Phi)$, associated by S. Fomin and A. Zelevinsky to a finite root system $\Phi$, count elements of the lattice \nc of noncrossing partitions of corresponding type by rank. Similar interpretations for the $h$-vector of the positive part of $\Delta (\Phi)$ are provided. The proof utilizes the appearance of the complex $\Delta (\Phi)$ in the context of the lattice \nc, in recent work of two of the authors, as well as an explicit shelling of $\Delta (\Phi)$.Comment: 20 pages, 1 figur

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

Cyclic sieving and cluster multicomplexes

Reiner, Stanton, and White \cite{RSWCSP} proved results regarding the enumeration of polygon dissections up to rotational symmetry. Eu and Fu \cite{EuFu} generalized these results to Cartan-Killing types other than A by means of actions of deformed Coxeter elements on cluster complexes of Fomin and Zelevinsky \cite{FZY}. The Reiner-Stanton-White and Eu-Fu results were proven using direct counting arguments. We give representation theoretic proofs of closely related results using the notion of noncrossing and semi-noncrossing tableaux due to Pylyavskyy \cite{PN} as well as some geometric realizations of finite type cluster algebras due to Fomin and Zelevinsky \cite{FZClusterII}.Comment: To appear in Adv. Appl. Mat