Related papers: Infinite Reduced Words, Lattice Property And Braid…

For an infinite Coxeter system, one can extend the weak right order to the set of infinite reduced words. This is called limit weak order. In [Transformation Groups 18(1), 2013, 179-231], Lam and Pylyavskyy showed that for affine Weyl…

In this paper, we initiate the study of the twisted weak order associated to a twisted Bruhat order for a Coxeter group and explore the relationship between the lattice property of such order and the infinite reduced words. We show that for…

We describe an algorithm to identify a minimal set of "braid relations" which span and preserve all sets of involution words for twisted Coxeter systems of finite or affine type. We classify the cases in which adding the smallest possible…

Extended affine Weyl groups are the Weyl groups of extended affine root systems. Finite presentations for extended affine Weyl groups are known only for nullities $\leq 2$, where for nullity 2 there is only one known such presentation. We…

We investigate in detail relationships between the set ${\mathfrak B}^\infty$ of all infinite ``biconvex'' sets in the positive root system $\Delta_+$ of an arbitrary untwisted affine Lie algebra ${\mathfrak g}$ and the set ${\mathcal…

In this paper, we investigate various properties of strong and weak twisted Bruhat orders on a Coxeter group. In particular, we prove that any twisted strong Bruhat order on an affine Weyl group is locally finite, strengthening a result of…

Let $W$ be a Coxeter group and let $\Phi^+$ be its positive roots. A subset $B$ of $\Phi^+$ is called biclosed if, whenever we have roots $\alpha$, $\beta$ and $\gamma$ with $\gamma \in \mathbb{R}_{>0} \alpha + \mathbb{R}_{>0} \beta$, if…

We compute all sections of the finite Weyl group, that satisfy the braid relations, in the case that G is an almost-simple connected reductive group defined over an algebraically closed field. We then demonstrate that this set of sections…

We lift the lattice of translations in the extended affine Weyl group to a braid group action on the quantum affine algebra. This action fixes the Heisenberg subalgebra pointwise. Loop like generators are found for the algebra which satisfy…

This is the second in a series of papers developing a theory of total positivity for loop groups. In this paper, we study infinite products of Chevalley generators. We show that the combinatorics of infinite reduced words underlies the…

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…

We attach to every Coxeter system (W,S) an extension C_W of the corresponding Iwahori-Hecke algebra. We construct a 1-parameter family of (generically surjective) morphisms from the group algebra of the corresponding Artin group onto C_W.…

This paper studies connections between the preprojective representations of a valued quiver, the (+)-admissible sequences of vertices, and the Weyl group by associating to each preprojective representation a canonical (+)-admissible…

Using previous results concerning the rank two and rank three cases, all connected simply connected Cartan schemes for which the real roots form a finite irreducible root system of arbitrary rank are determined. As a consequence one obtains…

The limit weak order on an affine Weyl group was introduced by Lam and Pylyavskyy in their study of total positivity for loop groups. They showed that in the case of the affine symmetric group the minimal elements of this poset coincide…

Let (W,S) be a finite rank Coxeter system with W infinite. We prove that the limit weak order on the blocks of infinite reduced words of W is encoded by the topology of the Tits boundary of the Davis complex X of W. We consider many special…

A classic result of Conway and Coxeter on frieze patterns has been generalized to a bijection between $p$-angulations of regular polygons and frieze patterns of type $\Lambda_p$. One of the features of Conway-Coxeter theory is a…

For W a finite (2-)reflection group and B its (generalized) braid group, we determine the Zariski closure of the image of B inside the corresponding Iwahori-Hecke algebra. The Lie algebra of this closure is reductive and generated in the…

We discuss the classification of reflection subgroups of finite and affine Weyl groups from the point of view of their root systems. A short case free proof is given of the well known classification of the isomorphism classes of reflection…

This paper contains a complete description of minimal non-gatherable triangle triples in the lambda-sequences for the affine classical root systems and some claims for arbitrary (reduced) affine root systems. It continues our previous paper…