Related papers: Cells and cacti
Cactus groups Jn are currently attracting considerable interest from diverse mathematical communities. This work explores their relations to right-angled Coxeter groups, and in particular twin groups Twn and Mostovoy's Gauss diagram groups…
In this article, we propose to initiate the general study of involution systems. An {\em involution system}, that is, a group $W$ generated by a set of involutions $S$, is naturally endowed with a {\em weak order} arising from orienting the…
The $\gamma$-Cambrian semilattices $\mathcal{C}_{\gamma}$ defined by Reading and Speyer are a family of meet-semilattices associated with a Coxeter group $W$ and a Coxeter element $\gamma\in W$, and they are lattices if and only if $W$ is…
Any function can be constructed using a hierarchy of simpler functions through compositions. Such a hierarchy can be characterized by a binary rooted tree. Each node of this tree is associated with a function which takes as inputs two…
We compute Aut(W) for any even Coxeter group whose Coxeter diagram is connected, contains no edges labeled 2, and cannot be separated into more than 2 connected components by removing a single vertex. The description is given explicitly in…
Motivated by the second author's construction of a classifying space for the group of pure symmetric automorphisms of a free product, we introduce and study a family of topological operads, the operads of based cacti, defined for every…
Cactus groups are traditionally defined based on symmetric groups, and pure cactus groups are particular subgroups of cactus groups. Mostovoy showed that pure cactus groups embed into right-angled Coxeter groups. We generalize this result…
In decision theory an act is a function from a set of conditions to the set of real numbers. The set of conditions is a partition in some algebra of events. The expected value of an act can be calculated when a probability measure is given.…
Given a bicategory C and a family W of arrows of C, we give conditions on the pair (C,W) that allow us to construct the bicategorical localization with respect to W by dealing only with the 2-cells, that is without adding objects or arrows…
A Coxeter group W is called reflection independent if its reflections are uniquely determined by W only, independently on the choice of the generating set. We give a new sufficient condition for the reflection independence, and examine this…
Let g be a Lie algebra of type A,D,E with fixed Cartan subalgebra h, root system R and Weyl group W. We show that a choice of Coxeter element C gives a root basis for g. Moreover we show that this root basis gives a purely combinatorial…
Let W be an arbitrary Coxeter group. If two elements have expressions that are cyclic shifts of each other (as words), then they are conjugate (as group elements) in W. We say that w is "cyclically fully commutative" (CFC) if every cyclic…
We outline an algorithm to recover the canonical (or, coarsest) stratification of a given finite-dimensional regular CW complex into cohomology manifolds, each of which is a union of cells. The construction proceeds by iteratively…
We determine a fundamental domain for the diagonal action of a finite Coxeter group $W$ on $V^{\oplus n}$, where $V$ is the reflection representation. This is used to give a stratification of $V^{\oplus n}$, which is respected by the group…
We propose a novel approach for studying rooted trees by using functions that we will call descent functions. We provide a construction method for rooted trees that allows to study their properties through the use of descent functions.…
We present families of combinatorial classes described as trees with nodes that can carry one of two types of "flowers": integer partitions or integer compositions. Two parameters on the flowers of trees will be considered: the number of…
We classify a class of complex representations of an arbitrary Coxeter group via characters of the integral homology of certain graphs. Such representations can be viewed as a generalization of the geometric representation and correspond to…
When W is a finite Coxeter group of classical type (A, B, or D), noncrossing partitions associated to W and compatibility of almost positive roots in the associated root system are known to be modeled by certain planar diagrams. We show how…
We construct, using finitely many generating cell and relations, props in the category of CW-complexes with the property that their associated operads are models for the $E_\infty$-operad. We use one of these to construct a cellular…
Connection of the Four Color Theorem (FCT) with some operations on trees is described. L.H. Kauffman's theorem about FCT and vector cross product is discussed. Operation of transplantation on trees linked with the move of brackets according…