Related papers: An atomic Coxeter presentation

For a Coxeter system (G,S) the multi-parametric alternating subalgebra H^+(G) of the Hecke algebra and the alternating subgroup B^+(G) of the braid group are defined. Two presentations for H^+(G) and B^+(G) are given; one generalizes the…

Let B be the generalized braid group associated to some finite complex reflection group. We define a representation of B of dimension the number of reflections of the corresponding reflection group, which generalizes the Krammer…

Involution words are variations of reduced words for twisted involutions in Coxeter groups. They arise naturally in the study of the Bruhat order, of certain Iwahori-Hecke algebra modules, and of orbit closures in flag varieties.…

Collisions of Bose-Einstein condensates can be used as a mean to generate correlated pairs of atoms. The scattered massive particles, in analogy to photon pairs in quantum optics, might be used in the violation of Bell's inequalities,…

We define and study an action of the symmetric group on the Yokonuma--Hecke algebra. This leads to the definition of two classes of algebras. The first one is connected with the image of the algebra of the braid group inside the…

We introduce bijections between generalized type $A_n$ noncrossing partitions (that is, associated to arbitrary standard Coxeter elements) and fully commutative elements of the same type. The latter index the diagram basis of the classical…

This paper gives a new, simplified presentation of the classical pure braid group. The generators are given by the squares of the longest elements over connected subgraphs, and we prove that the only relations are either commutators or…

The nilCoxeter algebra $\mathcal{N}S_n$ of the symmetric group $S_n$ is the algebra over $\mathbb{Z}$ with generators $Y_i$ ($1\leqslant i\leqslant n-1$), satisfying the braid relations $Y_iY_{i+1}Y_i=Y_{i+1}Y_iY_{i+1}$, $Y_iY_j=Y_jY_i$…

We introduce two order relations on finite Coxeter groups which refine the absolute and the Bruhat order, and establish some of their main properties. In particular we study the restriction of these orders to noncrossing partitions and show…

This paper provides some evidence for conjectural relations between extensions of (right) weak order on Coxeter groups, closure operators on root systems, and Bruhat order. The conjecture focused upon here refines an earlier question as to…

We demonstrate how a simple linear-algebraic technique used earlier to compute low-degree cohomology of current Lie algebras, can be utilized to compute other kinds of structures on such Lie algebras, and discuss further generalizations,…

We introduce Poisson double algebroids, and the equivalent concept of double Lie bialgebroid, which arise as second-order infinitesimal counterparts of Poisson double groupoids. We develop their underlying Lie theory, showing how these…

We extend the weak Bruhat order of a finite Coxeter group to the set of its coclasses, modulo parabolic standard subgroups. We use this order to describe associative algebra structures on the vector spaces spanned by the faces of…

The Demazure operator associated to a simple reflection satisfies the twisted Leibniz rule. In this paper we introduce a generalization of the twisted Leibniz rule for the Demazure operator associated to any atomic double coset. We prove…

In this paper, we study the structure of braid graphs in simply-laced Coxeter systems. We prove that every reduced expression has a unique factorization as a product of so-called links, which in turn induces a decomposition of the braid…

We define new deformations of group algebras of Coxeter groups W and of subgroups of even elements in them, by deforming the braid relations. We show that these deformations are algebraically flat iff they are formally flat, and that this…

There is a natural analogue of weak Bruhat order on the involutions in any Coxeter group. The saturated chains of intervals in this order correspond to reduced words for a certain set of group elements called atoms. Brion gives a general…

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…

Aiming for a revival of the theory of crystallographic complex reflection groups, we compute (minimal) Coxeter-like reflection presentations for the infinite families of those non-genuine groups which satisfy Steinberg's fixed point…

We introduce the relative Coxeter groupoid and construct intrinsic relative braid group symmetries for quantum supersymmetric pairs of type sAIII. These symmetries are constructed by establishing new intertwining properties of quasi…