Related papers: Groupoids, root systems and weak order II

This paper investigates the question of uniqueness of the reduced oriented matroid structure arising from root systems of a Coxeter group in real vector spaces. We settle the question for finite Coxeter groups, irreducible affine Weyl…

In this monograph we lay the foundation for a theory of coarse groups and coarse actions. Coarse groups are group objects in the category of coarse spaces, and can be thought of as sets with operations that satisfy the group axioms "up to…

We give an explicit handy (and cocycle-free) description of the groupoid of weak maps between two crossed-modules in terms of certain digrams of groups which we we call a {\em butterflies}. We define composition of butterflies and this way…

We translate the axioms of a Weyl groupoid with (not necessarily finite) root system in terms of arrangements. The result is a correspondence between Weyl groupoids permitting a root system and Tits arrangements satisfying an integrality…

A groupoid is a small category in which each morphism has an inverse. A topological groupoid is a groupoid in which both sets of objects and morphisms have topologies such that all groupoid structure maps are continuous. The notion of…

We survey the general theory of groupoids, groupoid actions, groupoid principal bundles, and various kinds of morphisms between groupoids in the framework of categories with pretopology. We study extra assumptions on pretopologies that are…

Presentations of smooth symmetry groups of differentiable stacks are studied within the framework of the weak 2-category of Lie groupoids, smooth principal bibundles, and smooth biequivariant maps. It is shown that principality of bibundles…

We first prove the Grinberg-Kazhdan formal arc theorem without any assumptions on the characteristic. This part of the article is equivalent to arXiv:math-AG/0203263. Then we try to clarify the geometric ideas behind the proof by…

For any type of fundamental groupoid scheme, we construct an algebraic cohomology theory for varieties with coefficients in the base field. This is a minor variant of \'etale cohomology, involving neither de Rham complexes nor…

We generalize the notion of a root system by relaxing the conditions that ensure that it is invariant under reflections and study the resulting structures, which we call generalized root systems (GRSs for short). Since both Kostant root…

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…

Let $(W,S)$ be an arbitrary Coxeter system. For each word $\omega$ in the generators we define a partial order--called the {\sf $\omega$-sorting order}--on the set of group elements $W_\omega\subseteq W$ that occur as subwords of $\omega$.…

We consider three forms of composition of matroids, each of which extends the category of bimatroids to a rigid monoidal category. Many well-known constructions are functorial or defined by morphisms in these categories. Motivating examples…

Motivated by work of Kac and Lusztig, we define a root system and a Weyl groupoid for a large class of semisimple Yetter-Drinfeld modules over an arbitrary Hopf algebra. The obtained combinatorial structure fits perfectly into an existing…

If $x$ and $y$ are roots in the root system with respect to the standard (Tits) geometric realization of a Coxeter group $W$, we say that $x$ \emph{dominates} $y$ if for all $w\in W$, $wy$ is a negative root whenever $wx$ is a negative…

Root systems are sets with remarkable symmetries and therefore they appear in many situations in mathematics. Among others, denominator formulae of root systems are very beautiful and mysterious equations which have several meanings from a…

We characterize those countable rooted trees whose full automorphism group has uncountable strong cofinality or contains an open subgroup with ample generics.

Graphs which generalize the simple or affine Dynkin diagrams are introduced. Each diagram defines a bilinear form on a root system and thus a reflection group. We present some properties of these groups and of their natural "Coxeter…

We look at Poisson geometry taking the viewpoint of singular foliations, understood as suitable submodules generated by Hamiltonian vector fields rather than partitions into (symplectic) leaves. The class of Poisson structures which behave…

We show that the order dimension of the weak order on a Coxeter group of type A, B or D is equal to the rank of the Coxeter group, and give bounds on the order dimensions for the other finite types. This result arises from a unified…