scholarly journals Treillis et bases des groupes de Coxeter

10.37236/1285 ◽  
1996 ◽  
Vol 3 (2) ◽  
Author(s):  
Alain Lascoux ◽  
Marcel-Paul Schützenberger
Keyword(s):  
Coxeter Groups ◽  
Bruhat Order ◽  
Ordered Sets ◽  
Finite Lattices

Finite lattices possess a basis, as well as a cobasis, from which the elements of the lattice can be recovered by sup or inf. We extend this construction to finite ordered sets, and then apply it to Coxeter groups, considered as ordered sets (Bruhat order). This amounts to embedding Coxeter groups into their enveloping lattices. These lattices are distributive in the cases of types An and Bn.

scholarly journals Tight Quotients and Double Quotients in the Bruhat Order

10.37236/1871 ◽  
2005 ◽  
Vol 11 (2) ◽  
Author(s):  
John R. Stembridge
Keyword(s):  
Coxeter Groups ◽  
Upper Bounds ◽  
Bruhat Order ◽  
Weyl Groups ◽  
Order Dimension ◽  
Parabolic Subgroups ◽  
Affine Weyl Groups ◽  
Positive Side ◽  
Maximal Parabolic Subgroups

It is a well-known theorem of Deodhar that the Bruhat ordering of a Coxeter group is the conjunction of its projections onto quotients by maximal parabolic subgroups. Similarly, the Bruhat order is also the conjunction of a larger number of simpler quotients obtained by projecting onto two-sided (i.e., "double") quotients by pairs of maximal parabolic subgroups. Each one-sided quotient may be represented as an orbit in the reflection representation, and each double quotient corresponds to the portion of an orbit on the positive side of certain hyperplanes. In some cases, these orbit representations are "tight" in the sense that the root system induces an ordering on the orbit that yields effective coordinates for the Bruhat order, and hence also provides upper bounds for the order dimension. In this paper, we (1) provide a general characterization of tightness for one-sided quotients, (2) classify all tight one-sided quotients of finite Coxeter groups, and (3) classify all tight double quotients of affine Weyl groups.

scholarly journals On the Weak Order of Coxeter Groups

2019 ◽  
Vol 71 (2) ◽  
pp. 299-336 ◽  
Author(s):  
Matthew Dyer
Keyword(s):  
Complete Lattice ◽  
Coxeter Groups ◽  
Closure Operator ◽  
Root Systems ◽  
Bruhat Order ◽  
Weak Order ◽  
Maximal Subgroups ◽  
Rank Zero ◽  
Power Set ◽  
Rank One

AbstractThis 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 whether the set of initial sections of reflection orders, ordered by inclusion, forms a complete lattice. Meet and join in weak order are described in terms of a suitable closure operator. Galois connections are defined from the power set of $W$ to itself, under which maximal subgroups of certain groupoids correspond to certain complete meet subsemilattices of weak order. An analogue of weak order for standard parabolic subsets of any rank of the root system is defined, reducing to the usual weak order in rank zero, and having some analogous properties in rank one (and conjecturally in general).

scholarly journals SB-Labelings, Distributivity, and Bruhat Order on Sortable Elements

10.37236/4942 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Henri Mühle
Keyword(s):  
Coxeter Group ◽  
Coxeter Groups ◽  
Bruhat Order ◽  
Distributive Lattices ◽  
Coxeter Element ◽  
General Statement ◽  
Coxeter Diagram

In this article, we investigate the set of $\gamma$-sortable elements, associated with a Coxeter group $W$ and a Coxeter element $\gamma\in W$, under Bruhat order, and we denote this poset by $\mathcal{B}_{\gamma}$. We show that this poset belongs to the class of SB-lattices recently introduced by Hersh and Mészáros, by proving a more general statement, namely that all join-distributive lattices are SB-lattices. The observation that $\mathcal{B}_{\gamma}$ is join-distributive is due to Armstrong. Subsequently, we investigate for which finite Coxeter groups $W$ and which Coxeter elements $\gamma\in W$ the lattice $\mathcal{B}_{\gamma}$ is in fact distributive. It turns out that this is the case for the "coincidental" Coxeter groups, namely the groups $A_{n},B_{n},H_{3}$ and $I_{2}(k)$. We conclude this article with a conjectural characteriziation of the Coxeter elements $\gamma$ of said groups for which $\mathcal{B}_{\gamma}$ is distributive in terms of forbidden orientations of the Coxeter diagram.

scholarly journals Bruhat order of Coxeter groups and shellability

1982 ◽  
Vol 43 (1) ◽  
pp. 87-100 ◽  
Author(s):  
Anders Björner ◽  
Michelle Wachs
Keyword(s):  
Coxeter Groups ◽  
Bruhat Order

scholarly journals Bruhat interval polytopes

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Emmanuel Tsukerman ◽  
Lauren Williams
Keyword(s):  
Coxeter Groups ◽  
Toda Lattice ◽  
Bruhat Order ◽  
Flag Variety ◽  
Toda Hierarchy ◽  
Dimension Formula ◽  
Sont Respectivement ◽  
International Audience ◽  
The Moment ◽  
The Relationship

International audience Let $u$ and $v$ be permutations on $n$ letters, with $u$ &le; $v$ in Bruhat order. A <i>Bruhat interval polytope</i> $Q_{u,v}$ is the convex hull of all permutation vectors $z=(z(1),z(2),...,z(n))$ with $u$ &le; $z$ &le; $v$. Note that when $u=e$ and $v=w_0$ are the shortest and longest elements of the symmetric group, $Q_{e,w_0}$ is the classical permutohedron. Bruhat interval polytopes were studied recently in the 2013 paper “The full Kostant-Toda hierarchy on the positive flag variety” by Kodama and the second author, in the context of the Toda lattice and the moment map on the flag variety. In this paper we study combinatorial aspects of Bruhat interval polytopes. For example, we give an inequality description and a dimension formula for Bruhat interval polytopes, and prove that every face of a Bruhat interval polytope is a Bruhat interval polytope. A key tool in the proof of the latter statement is a generalization of the well-known lifting property for Coxeter groups. Motivated by the relationship between the lifting property and $R$-polynomials, we also give a generalization of the standard recurrence for $R$-polynomials. Soient $u$ et $v$ des permutations sur $n$ lettres, avec, $u$ &le; $v$ dans l’ordre de Bruhat. Un <i>polytope d’intervalles de Bruhat</i> $Q_{u,v}$ est l’enveloppe convexe de tous les vecteurs de permutations $z=(z(1),z(2),...,z(n))$ avec $u$ &le; $z$ &le; $v$. Notons que lorsque $u=e$ et $v=w_0$ sont respectivement le plus court et le plus long élément du groupe symétrique, $Q_{e,w_0}$ est le permutoèdre classique. Les polytopes d’intervalles de Bruhat ont été étudiés récemment dans le papier de 2013 “The full Kostant-Toda hierarchy on the positive flag variety” par Kodama et le deuxième auteur, dans le contexte du treillis de Toda et la carte des moments sur la variété de drapeaux. Dans ce papier nous étudions des aspects combinatoires des polytopes d’intervalles de Bruhat. Par exemple, nous donnons une description par inégalités et une formule dimensionnelle pour les polytopes d’intervalles de Bruhat, et prouvons que chaque face d’un polytope d’intervalles de Bruhat est un polytope d’intervalles de Bruhat. Un outil essentiel dans la preuve de cette dernière affirmation est une généralisation de la célèbre propriété de lifting pour les groupes de Coxeter. Motivés par la relation entre la propriété de lifting et les $R$-polynômes, nous donnons aussi une généralisation de la récurrence standard pour les $R$-polynômes.

scholarly journals An improved tableau criterion for Bruhat order

10.37236/1246 ◽  
1996 ◽  
Vol 3 (1) ◽  
Author(s):  
Anders Björner ◽  
Francesco Brenti
Keyword(s):  
Coxeter Groups ◽  
Bruhat Order

To decide whether two permutations are comparable in Bruhat order of $S_n$ with the well-known tableau criterion requires $\binom{n}{2}$ comparisons of entries in certain sorted arrays. We show that to decide whether $x\le y$ only $d_1+d_2+...+d_k$ of these comparisons are needed, where $\{d_1,d_2,...,d_k\} = \{i|x(i)>x(i+1)\}$. This is obtained as a consequence of a sharper version of Deodhar's criterion, which is valid for all Coxeter groups.

scholarly journals Higher Bruhat Orders in Type B

10.37236/5620 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Seth Shelley-Abrahamson ◽  
Suhas Vijaykumar
Keyword(s):  
Symmetric Group ◽  
Minimal Element ◽  
Partially Ordered Sets ◽  
Bruhat Order ◽  
Hyperplane Arrangements ◽  
Type B ◽  
Ordered Sets ◽  
Weak Bruhat Order ◽  
Positive Integers ◽  
Maximal Chains

Motivated by the geometry of hyperplane arrangements, Manin and Schechtman defined for each integer $n \geq 1$ a hierarchy of finite partially ordered sets $B(n, k),$ indexed by positive integers $k$, called the higher Bruhat orders.  The poset $B(n, 1)$ is naturally identified with the weak left Bruhat order on the symmetric group $S_n$, each $B(n, k)$ has a unique maximal and a unique minimal element, and the poset $B(n, k + 1)$ can be constructed from the set of maximal chains in $B(n, k)$.  Ben Elias has demonstrated a striking connection between the posets $B(n, k)$ for $k = 2$ and the diagrammatics of Bott-Samelson bimodules in type A, providing significant motivation for the development of an analogous theory of higher Bruhat orders in other Cartan-Killing types, particularly for $k = 2$.  In this paper we present a partial generalization to type B, complete up to $k = 2$, prove a direct analogue of the main theorem of Manin and Schechtman, and relate our construction to the weak Bruhat order and reduced expression graph for Weyl group $B_n$.

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