Tight quotients and double quotients in the Bruhat order.
type: Item
instance of: scholarly article
title: Tight quotients and double quotients in the Bruhat order.
zbMATH Open document ID: 1067.20053
author: John R. Stembridge
published in: The Electronic Journal of Combinatorics
publication date: 2005-04-25T00:00:00Z
full work available at URL: https://eudml.org/doc/124886
full work available at URL: v11i2r14.html
review text: Summary: 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.
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 06A07
zbMATH DE Number: 2161264
zbMATH Keywords: Bruhat order
zbMATH Keywords: Coxeter groups
zbMATH Keywords: maximal parabolic subgroups
zbMATH Keywords: reflection representations
zbMATH Keywords: root systems
zbMATH Keywords: order dimensions
zbMATH Keywords: affine Weyl groups
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2161264