A characterization of the simply-laced FC-finite Coxeter groups.
type: Item
instance of: scholarly article
title: A characterization of the simply-laced FC-finite Coxeter groups.
zbMATH Open document ID: 1070.20046
author: Manabu Hagiwara
author: Masao Ishikawa
author: Hiroyuki Tagawa
published in: Annals of Combinatorics
publication date: 2004-09-27T00:00:00Z
review text: Let \((W,S)\) be a Coxeter system. An element of \(W\) is called fully commutative element (or \(w\) is fully commutative) if any reduced expression of \(w\) can be converted into any other reduced expression of \(w\) by exchanging adjacent commuting generators several times. A Coxeter group is called `FC-finite' if the number of its fully commutative elements is finite. For \(s,t\in S\), one denotes the order of \(st\) by \(m(st)\). If \(\{m(st)\mid s,t\in S\}\subset\{1,2,3\}\), then one calls \((W,S)\) (resp. \(W\)) a simply-laced Coxeter system (resp. a simply-laced Coxeter group). For two elements \(w\) and \(x\) of \(W\), we say that \(w\) covers \(x\) if \(x<w\) and \(l(x)=l(w)-1\). We call a Coxeter group bi-full if its fully commutative elements coincide with its fully covering elements. In this paper the authors show that the bi-full Coxeter groups are the ones of type \(A_n\), \(D_n\), \(E_n\) with no restriction on \(n\). In other words, Coxeter groups of type \(E_9,E_{10},\dots\) are also bi-full. According to a result of \textit{C. K. Fan} [Ph.D. Thesis, MIT (1995)], a Coxeter group is a simple-laced FC-finite Coxeter group if and only if it is a bi-full Coxeter group.
reviewed by: Chen Chengdong
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 06A07
zbMATH DE Number: 2103722
zbMATH Keywords: Coxeter groups
zbMATH Keywords: Bruhat ordering
zbMATH Keywords: fully commutative elements
zbMATH Keywords: fully covering elements
MaRDI profile type: Publication
full work available at URL: s00026-004-0214-6
OpenAlex ID: W1973786555
DOI: 10.1007/S00026-004-0214-6
description: scientific article; zbMATH DE number 2103722