Related papers: The facial weak order on hyperplane arrangements

A facet of an hyperplane arrangement is called external if it belongs to exactly one bounded cell. The set of all external facets forms the envelope of the arrangement. The number of external facets of a simple arrangement defined by $n$…

Let ${\mathcal A}$ be a finite real linear hyperplane arrangement in three dimensions. Suppose further that all the regions of ${\mathcal A}$ are isometric. We prove that ${\mathcal A}$ is necessarily a Coxeter arrangement. As it is well…

Drawing parallels with hyperplane arrangements, we develop the theory of arrangements of submanifolds. Given a smooth, finite dimensional, real manifold $X$ we consider a finite collection $\mathcal{A}$ of locally flat, codimension-1…

A weak pseudoline arrangement is a topological generalization of a line arrangement, consisting of curves topologically equivalent to lines that cross each other at most once. We consider arrangements that are outerplanar---each crossing is…

Over any partially ordered abelian group whose positive cone is closed in an appropriate sense and has finitely many faces, modules that satisfy a weak finiteness condition admit finite primary decompositions. This conclusion rests on the…

We construct a poset from a simple acyclic digraph together with a valuation on its vertices, and we compute the values of its M\"obius function. We show that the weak order on Coxeter groups of type A, B, affine A, and the flag weak order…

In this article, we investigate the existence of joins in the weak order of an infinite Coxeter group W. We give a geometric characterization of the existence of a join for a subset X in W in terms of the inversion sets of its elements and…

Given a symmetric monoidal category $C$ with product $\sqcup$, where the neutral element for the product is an initial object, we consider the poset of $\sqcup$-complemented subobjects of a given object $X$. When this poset has finite…

Hypergraphic polytopes $\Delta_{\mathbb{H}}$ arise as Minkowski sums of simplices indexed by the hyperedges of a hypergraph $\mathbb{H}$. Orienting the $1$-skeleton of such a polytope by a certain generic linear functional gives rise to the…

We extend the homotopy theories based on point reduction for finite spaces and simplicial complexes to finite acyclic categories and $\Delta$-complexes, respectively. The functors of classifying spaces and face posets are compatible with…

This paper introduces the geometric foundations for the study of the $s$-permutahedron and the $s$-associahedron, two objects that encode the underlying geometric structure of the $s$-weak order and the $s$-Tamari lattice. We introduce the…

This paper investigates the problem of listing faces of combinatorial polytopes, such as hypercubes, permutahedra, associahedra, and their generalizations. Firstly, we consider the face lattice, which is the inclusion order of all faces of…

We give a necessary and sufficient condition for that the support $\tau$-tilting poset of a finite dimensional algebra is isomorphic to the poset of symmetric group with weak order. Moreover we show that there are infinitely many finite…

We show that a connected finite topological space with $12$ or less points has a weak homotopy type of a wedge of spheres. In other words, we show that the order complex of a connected finite poset with $12$ or less points has a homotopy…

We introduce in this paper a definition of (non necessarily positive) opetopes where faces are organised in a poset. Then we show that this description is equivalent to that given in terms of constellations by Kock, Joyal, Batanin and…

This paper shows how to construct a discrete Morse function with a relatively small number of critical cells for the order complex of any finite poset with $\hat{0} $ and $\hat{1}$ from any lexicographic order on its maximal chains.…

We study linear relations between face numbers of levels in arrangements. Let $V = \{ v_1, \ldots, v_n \} \subset \mathbf{R}^{r}$ be a vector configuration in general position, and let $\mathcal{A}(V)$ be polar dual arrangement of…

We study a class of polyhedra associated to marked posets. Examples of these polyhedra are Gelfand-Tsetlin polytopes and cones, as well as Berenstein-Zelevinsky polytopes, all of which have appeared in the representation theory of…

Given a hypersurface coamoeba of a Laurent polynomial f, it is an open problem to describe the structure of its set of connected complement components. In this paper we approach this problem by introducing the lopsided coamoeba. We show…

We develop the theory of arrangements of spheres. Consider a finite collection of codimension-$1$ subspheres in a positive-dimensional sphere. There are two posets associated with this collection: the poset of faces and the poset of…