Related papers: Removahedral congruences versus permutree congruen…
We study the problem of when the collection of the recession cones of a polyhedral complex forms also a complex. We exhibit an example showing that this is no always the case. We also show that if the support of the given polyhedral complex…
For all classical groups (and for their analogs in infinite dimension or over general base fields or rings) we construct certain contractions, called "homotopes". The construction is geometric, using as ingredient involutions of associative…
For all classical groups (and for their analogs in infinite dimension or over general base fields or rings) we construct certain contractions, called "homotopes". The construction is geometric, using as ingredient involutions of associative…
This is an investigation of the role of shuffling and concatenating in the theory of graph drawing. A simple syntactic description of these and related operations is proved complete in the context of finite partial orders, as general as…
Viewing a fan as a partially ordered set (of cones) we consider a category of sheaves on the fan which corresponds to a category of equivariant sheaves on the corresponding toric variety if the fan is rational. In this category we define an…
We develop a method for subdividing polyhedral complexes in a way that restricts the possible recession cones and allows one to work with a fixed class of polyhedron. We use these results to construct locally finite completions of rational…
The paper considers systems of contraction similarities in $\mathbb R^d$ sending a given polyhedron $P$ to polyhedra $P_i\subset P$, whose non-empty intersections are singletons and contain the common vertices of those polyhedra, while the…
In a finite-dimensional real vector space furnished with a rational structure with respect to a subfield of the field of real numbers, every (simplicial) rational semifan is contained in a complete (simplicial) rational semifan. In this…
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…
A graph associahedron is a simple polytope whose face lattice encodes the nested structure of the connected subgraphs of a given graph. In this paper, we study certain graph properties of the 1-skeleta of graph associahedra, such as their…
We introduce the notion of Lorentzian fans, which form a special class of tropical fans that are particularly well-suited for proving Alexandrov-Fenchel type inequalities. To demonstrate the utility of Lorentzian fans, we prove a practical…
It is shown that coherence conditions for monoidal categories concerning associativity are analogous to coherence conditions for symmetric or braided strictly monoidal categories, where associativity arrows are identities. Mac Lane's…
We describe the cone of deformations of a Coxeter permutahedron, or equivalently, the nef cone of the toric variety associated to a Coxeter complex. This family of polytopes contains polyhedral models for the Coxeter-theoretic analogs of…
We consider amplituhedron-like geometries which are defined in a similar way to the intrinsic definition of the amplituhedron but with non-maximal winding number. We propose that for the cases with minimal number of points the canonical…
A fan is an arcwise-connected continuum, which is hereditarily unicoherent and has exactly one ramification point. Many of the known examples of fans were constructed as 1-dimensional continua that are unions of arcs which intersect in…
We continue the approach toward a purely combinatorial "virtual" intersection cohomology for possibly non-rational fans, based on our investigation of equivariant intersection cohomology for toric varieties (see math.AG/9904159).…
Flow cones of a directed acyclic graph admit a family of unimodular triangulations given by Danilov, Karzanov, and Koshevoy (DKK) whose normal fans are related to (generalizations) of the associahedron and permutahedron. A correspondence…
We introduce two generalizations of bracket vectors from binary trees to permutrees. These new vectors help describe algebraic and geometric properties of the rotation lattice of permutrees defined by Pilaud and Pons. The first…
In this article we prove in the main theorem that, there is a bijection between the isomorphism classes of a certain type of real hyperplane arrangements on the one hand, and the antipodal pairs of convex cones of an associated…
For each poset $P$, we construct a polytope $A(P)$ called the $P$-associahedron. Similarly to the case of graph associahedra, the faces of $A(P)$ correspond to certain nested collections of subsets of $P$. The Stasheff associahedron is a…