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

Minkowski decompositions for generalized associahedra of acyclic type

We give an explicit subword complex description of the generators of the type cone of the g-vector fan of a finite type cluster algebra with acyclic initial seed. This yields in particular a description of the Newton polytopes of the F-polynomials in terms of subword complexes as conjectured by S. Brodsky and the third author. We then show that the cluster complex is combinatorially isomorphic to the totally positive part of the tropicalization of the cluster variety as conjectured by D. Speyer and L. Williams.Comment: 17 pages. v2: updated and extended examples, added footnote that Theorem 1.4 also follows from [AHL20, Theorems 4.1 & 4.2

Associahedra via spines

An associahedron is a polytope whose vertices correspond to triangulations of a convex polygon and whose edges correspond to flips between them. Using labeled polygons, C. Hohlweg and C. Lange constructed various realizations of the associahedron with relevant properties related to the symmetric group and the classical permutahedron. We introduce the spine of a triangulation as its dual tree together with a labeling and an orientation. This notion extends the classical understanding of the associahedron via binary trees, introduces a new perspective on C. Hohlweg and C. Lange's construction closer to J.-L. Loday's original approach, and sheds light upon the combinatorial and geometric properties of the resulting realizations of the associahedron. It also leads to noteworthy proofs which shorten and simplify previous approaches.Comment: 27 pages, 11 figures. Version 5: minor correction

Subword complexes, cluster complexes, and generalized multi-associahedra

In this paper, we use subword complexes to provide a uniform approach to finite type cluster complexes and multi-associahedra. We introduce, for any finite Coxeter group and any nonnegative integer k, a spherical subword complex called multi-cluster complex. For k=1, we show that this subword complex is isomorphic to the cluster complex of the given type. We show that multi-cluster complexes of types A and B coincide with known simplicial complexes, namely with the simplicial complexes of multi-triangulations and centrally symmetric multi-triangulations respectively. Furthermore, we show that the multi-cluster complex is universal in the sense that every spherical subword complex can be realized as a link of a face of the multi-cluster complex.Comment: 26 pages, 3 Tables, 2 Figures; final versio

Brick polytopes of spherical subword complexes and generalized associahedra

International audienceWe generalize the brick polytope of V. Pilaud and F. Santos to spherical subword complexes for finite Coxeter groups. This construction provides polytopal realizations for a certain class of subword complexes containing all cluster complexes of finite types. For the latter, the brick polytopes turn out to coincide with the known realizations of generalized associahedra, thus opening new perspectives on these constructions. This new approach yields in particular the vertex description of generalized associahedra, a Minkowski sum decomposition into Coxeter matroid polytopes, and a combinatorial description of the exchange matrix of any cluster in a finite type cluster algebra

Convex Geometry of Subword Complexes of Coxeter Groups

This monography presents results related to the convex geometry of a family of simplicial complexes called ``subword complexes''. These simplicial complexes are defined using the Bruhat order of Coxeter groups. Despite a simple combinatorial definition much of their combinatorial properties are still not understood. In contrast, many of their known connections make use of specific geometric realizations of these simplicial complexes. When such realizations are missing, many connections can only be conjectured to exist. This monography lays down a framework using an alliance of algebraic combinatorics and discrete geometry to study further subword complexes. It provides an abstract, though transparent, perspective on subword complexes based on linear algebra and combinatorics on words. The main contribution is the presentation of a universal partial oriented matroid whose realizability over the real numbers implies the realizability of subword complexes as oriented matroids.Diese Monographie präsentiert Ergebnisse im Zusammenhang mit einer Familie von simplizialen Komplexen, die "Subwortkomplexe" genannt werden. Diese Simplizialkomplexe werden mit Hilfe der Bruhat-Ordnung von Coxeter-Gruppen definiert. Trotz einer einfachen kombinatorischen Definition werden viele ihrer kombinatorischen Eigenschaften immer noch nicht verstanden. Spezifische geometrische Realisierungen dieser Simplizialkomplexe machen neue Herangehensweisen an Vermutungen des Gebiets m\"oglich. Wenn solche Verbindungen fehlen, können viele Zusammenhänge nur vermutet werden. Diese Monographie legt einen Rahmen fest, in dem eine Allianz aus algebraischer Kombinatorik und diskreter Geometrie verwendet wird, um weitere Subwortkomplexe zu untersuchen. Es bietet eine abstrakte und transparente Perspektive auf Teilwortkomplexe, die auf linearer Algebra und Kombinatorik von Wörtern basiert. Der Hauptbeitrag ist die Darstellung eines universellen, nur teilweise orientierten Matroids, dessen Realisierbarkeit über den reellen Zahlen die Realisierbarkeit von Teilwortkomplexen als orientierte Matroide impliziert