相关论文: Minuscule Coxeter Dressians

We introduce partially ordered sets (posets) with an additional structure given by a collection of vector subspaces of an algebra $A$. We call them algebraically equipped posets. Some particular cases of these, are generalized equipped…

We introduce a generalization of tropical polyhedra able to express both strict and non-strict inequalities. Such inequalities are handled by means of a semiring of germs (encoding infinitesimal perturbations). We develop a tropical…

Given a graph G, we construct a simple, convex polytope whose face poset is based on the connected subgraphs of G. This provides a natural generalization of the Stasheff associahedron and the Bott-Taubes cyclohedron. Moreover, we show that…

The purpose of this paper is to study stable representations of partially ordered sets (posets) and compare it to the well known theory for quivers. In particular, we prove that every indecomposable representation of a poset of finite type…

In this paper we give a short, elementary proof of a known result in tropical mathematics, by which the convexity of the column span of a zero--diagonal real matrix $A$ is characterized by $A$ being a Kleene star. We give applications to…

We define a class of partial orders on a Coxeter group associated with sets of reflections. In special cases, these lie between the left weak order and the Bruhat order. We prove that these posets are graded by the length function and that…

We introduce tropical analogues of the notion of volume of polytopes, leading to a tropical version of the (discrete) classical isoperimetric inequality. The planar case is elementary, but a higher-dimensional generalization leads to an…

We introduce a notion of essential hyperbolic Coxeter polytope as a polytope which fits some minimality conditions. The problem of classification of hyperbolic reflection groups can be easily reduced to classification of essential Coxeter…

This survey is about combinatorial objects related to reflection groups and their applications in representation theory and arithmetic geometry. Coxeter groups and folded galleries in Coxeter complexes are introduced in detail and…

In the first section we discuss Morita invariance of differentiable/algebroid cohomology. In the second section we present an extension of the van Est isomorphism to groupoids. This immediately implies a version of Haefliger's conjecture…

This is a survey on tropical polytopes from the combinatorial point of view and with a focus on algorithms. Tropical convexity is interesting because it relates a number of combinatorial concepts including ordinary convexity, monomial…

We introduce a sheaf-theoretic approach to tropical homology, especially for tropical homology with potentially non-compact supports. Our setup is suited to study the functorial properties of tropical homology, and we show that it behaves…

In this paper, we give an explicit description of tropical cohomology of smooth algebraic varieties over trivially valued fields. We also construct ``monodromy weight'' spectral sequences for tropical cohomology of geometric strictly…

We study the fan structure of Dressians $\Dr(d,n)$ and local Dressians $\Dr(\cM)$ for a given matroid $\cM$. In particular we show that the fan structure on $\Dr(\cM)$ given by the three term Pl\"ucker relations coincides with the structure…

A traditional wavelet is a special case of a vector in a separable Hilbert space that generates a basis under the action of a system of unitary operators defined in terms of translation and dilation operations. A Coxeter/fractal-surface…

It is well known that a compact two dimensional surface is homeomorphic to a polygon with the edges identified in pairs. This paper not only presents a new proof of this statement but also generalizes it to any connected $n$-dimensional…

This paper investigates the combinatorics that gives rise to the Boltzmann probability distribution. Despite being one of the most important distributions in physics and other fields of science, the mathematics of the underlying model of…

We utilize the coherent-constructible correspondence to construct full strongly exceptional collections of nef line bundles in the derived category of a toric variety through the combinatorics of constructible sheaves built from polytopes.…

We consider a bivariate polynomial that generalizes both the length and reflection length generating functions in a finite Coxeter group. In seeking a combinatorial description of the coefficients, we are led to the study of a new Mahonian…

We present an unexpected application of tropical convexity to the determination of invariants for linear systems of differential equations. We show that the classical G\'erard-Levelt lattice saturation procedure can be geometrically…