Related papers: Local Finiteness of the Twisted Bruhat Orders on A…

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…

Coxeter groups are equipped with a partial order known as the weak order, such that $u \leq v$ if the inversions of $u$ are a subset of the inversions of $v$. In finite Coxeter groups, weak order is a complete lattice, but in infinite…

We classify cocovers and covers of a given element of the double affine Weyl semigroup W with respect to the Bruhat order, specifically when W is associated to a finite root system that is irreducible and simply laced. We show two…

We define a natural lattice structure on all subsets of a finite root system that extends the weak order on the elements of the corresponding Coxeter group. For crystallographic root systems, we show that the subposet of this lattice…

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…

We introduce the annex of an element $x$ in a Coxeter group as the set of elements $y$ such that $x \nleq y$ with respect to Bruhat order. This notion provides a complementary perspective to the study of Bruhat intervals and their…

In this paper, we establish a bijection between the infinite reduced words of an affine Weyl group and certain biclosed sets of its positive system and determine all finitely generated biclosed sets in the positive system of an affine Weyl…

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…

In 2011, Dyer published a series of conjectures on the weak order of Coxeter groups. One of these conjectures stated that the inversion set of the join of two elements in a Coxeter group is equal to some "closure" of the union of their…

We extend properties of the weak order on finite Coxeter groups to Weyl groupoids admitting a finite root system. In particular, we determine the topological structure of intervals with respect to weak order, and show that the set of…

We prove that the weak order on an infinite Coxeter group contains infinite antichains if and only if the group is not affine.

In this article we provide a new finite class of elements in any Coxeter system (W,S) called low elements. They are defined from Brink and Howlett's small roots, which are strongly linked to the automatic structure of (W,S). Our first main…

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…

The weak order is a classical poset structure on a Coxeter group; it is a lattice when the group is finite but merely a meet-semilattice when the group is infinite. Motivated by problems in Kazhdan--Lusztig theory, Matthew Dyer introduced…

Let $G$ be a split Kac-Moody group over a local field. In their study of the Iwahori-Hecke algebra of $G$, A.Braverman, D. Kazhdan and M. Patnaik defined a partial order - called the affine Bruhat order - on the extended affine Weyl…

Let (W, S) be a Coxeter system. We investigate combinatorially certain partial orders, called extended Bruhat orders, on a (W x W)-set W(N,C), which depends on W, a subset N of S, and a component C of N. We determine the length of the…

We investigate the ways in which fundamental properties of the weak Bruhat order on a Weyl group can be lifted (or not) to a corresponding highest weight crystal graph, viewed as a partially ordered set; the latter projects to the weak…

We study the restriction of the strong Bruhat order on an arbitrary Coxeter group $W$ to cosets $x W_L^\theta$, where $x$ is an element of $W$ and $W_L^\theta$ the subgroup of fixed points of an automorphism $\theta$ of order at most two of…

Given a Coxeter system (W,S) equipped with an involutive automorphism T, the set of twisted identities is i(T) = {T(w)^{-1}w : w \in W}. We point out how i(T) shows up in several contexts and prove that if there is no s \in S such that…

The Demazure product gives a natural monoid structure on any Coxeter group. Such structure occurs naturally in many different areas in Lie Theory. This paper studies the Demazure product of an extended affine Weyl group $\tilde W$. The main…