scholarly journals The Complete cd-Index of Dihedral and Universal Coxeter Groups

10.37236/661 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Saúl A. Blanco
Keyword(s):  
Coxeter Groups ◽  
Bruhat Graph ◽  
Coxeter Systems ◽  
Bruhat Intervals

We present a description, including a characterization, of the complete cd-index of dihedral intervals. Furthermore, we describe a method to compute the complete cd-index of intervals in universal Coxeter groups. To obtain such descriptions, we consider Bruhat intervals for which Björner and Wachs's CL-labeling can be extended to paths of different lengths in the Bruhat graph. While such an extension cannot be defined for all Bruhat intervals, it can be in dihedral and universal Coxeter systems.

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 Weak Generalized Lifting Property, Bruhat Intervals, and Coxeter Matroids

2020 ◽  
Author(s):  
Fabrizio Caselli ◽  
Michele D’Adderio ◽  
Mario Marietti
Keyword(s):  
Coxeter Group ◽  
Coxeter Groups ◽  
Combinatorial Properties ◽  
Coxeter Matroid ◽  
Finite Coxeter Group ◽  
Bruhat Intervals

Abstract We provide a weaker version of the generalized lifting property that holds in complete generality for all Coxeter groups, and we use it to show that every parabolic Bruhat interval of a finite Coxeter group is a Coxeter matroid. We also describe some combinatorial properties of the associated polytopes.

scholarly journals Depth in Coxeter groups of type $B$

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Eli Bagno ◽  
Riccardo Biagioli ◽  
Mordechai Novick
Keyword(s):  
Coxeter Group ◽  
Simple Formula ◽  
Coxeter Groups ◽  
Type B ◽  
Minimal Path ◽  
Signed Permutation ◽  
Signed Permutations ◽  
International Audience ◽  
Path Cost

International audience The depth statistic was defined for every Coxeter group in terms of factorizations of its elements into product of reflections. Essentially, the depth gives the minimal path cost in the Bruaht graph, where the edges have prescribed weights. We present an algorithm for calculating the depth of a signed permutation which yields a simple formula for this statistic. We use our algorithm to characterize signed permutations having depth equal to length. These are the fully commutative top-and-bottom elements defined by Stembridge. We finally give a characterization of the signed permutations in which the reflection length coincides with both the depth and the length. La statistique profondeur a été introduite par Petersen et Tenner pour tout groupe de Coxeter $W$. Elle est définie pour tout $w \in W$ à partir de ses factorisations en produit de réflexions (non nécessairement simples). Pour le type $B$, nous introduisons un algorithme calculant la profondeur, et donnant une formule explicite pour cette statistique. On utilise par ailleurs cet algorithme pour caractériser tous les éléments ayant une profondeur égale à leur longueur. Ces derniers s’avèrent être les éléments pleinement commutatifs “hauts-et-bas” introduits par Stembridge. Nous donnons enfin une caractérisation des éléments dont la longueur absolue, la profondeur et la longueur coïncident.

scholarly journals Flip Posets of Bruhat Intervals

10.37236/7910 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Saúl A. Blanco
Keyword(s):  
Maximal Chain ◽  
Combinatorial Description ◽  
Bruhat Graph ◽  
Eulerian Posets ◽  
Bruhat Intervals

In this paper we introduce a way of partitioning the paths of shortest lengths in the Bruhat graph $B(u,v)$ of a Bruhat interval $[u,v]$ into rank posets $P_{i}$ in a way that each $P_{i}$ has a unique maximal chain that is rising under a reflection order. In the case where each $P_{i}$ has rank three, the construction yields a combinatorial description of some terms of the complete $\textbf{cd}$-index as a sum of ordinary $\textbf{cd}$-indices of Eulerian posets obtained from each of the $P_{i}$.

scholarly journals Shortest path poset of finite Coxeter groups

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Saúl A. Blanco
Keyword(s):  
Shortest Path ◽  
Coxeter Group ◽  
Coxeter Groups ◽  
Shortest Paths ◽  
Combinatorial Interpretation ◽  
Nous Permet ◽  
Finite Coxeter Group ◽  
International Audience ◽  
Absolute Length ◽  
Bruhat Graph

International audience We define a poset using the shortest paths in the Bruhat graph of a finite Coxeter group $W$ from the identity to the longest word in $W, w_0$. We show that this poset is the union of Boolean posets of rank absolute length of $w_0$; that is, any shortest path labeled by reflections $t_1,\ldots,t_m$ is fully commutative. This allows us to give a combinatorial interpretation to the lowest-degree terms in the complete $\textbf{cd}$-index of $W$. Nous définissons un poset en utilisant le plus court chemin entre l'identité et le plus long mot de $W, w_0$, dans le graph de Bruhat du groupe finie Coxeter, $W$. Nous prouvons que ce poset est l'union de posets Boolean du même rang que la longueur absolute de $w_0$; ça signifie que tous les plus courts chemins, étiquetés par réflexions $t_1,\ldots, t_m$ sont totalement commutatives. Ça nous permet de donner une interprétation combinatoire aux termes avec le moindre grade dans le $\textbf{cd}$-index complet de $W$.

A Characterization of the Simply-Laced FC-Finite Coxeter Groups

2004 ◽  
Vol 8 (2) ◽  
pp. 177-196 ◽  
Author(s):  
Manabu Hagiwara ◽  
Masao Ishikawa ◽  
Hiroyuki Tagawa
Keyword(s):  
Coxeter Groups
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