Upper bounds in affine Weyl groups under the weak order
type: Item
instance of: scholarly article
title: Upper bounds in affine Weyl groups under the weak order
zbMATH Open document ID: 0952.20031
DOI: 10.1023/A:1006352408688
author: Debra J. Waugh
published in: Order
publication date: 2000-10-17T00:00:00Z
review text: Let \((W,S)\) be a finitely generated Coxeter group. For \(w=s_1\cdots s_k\in W\), with \(s_i\in S\) for all \(i\) and \(k\) minimal, we say that \(s_1\cdots s_k\) is a reduced expression for \(w\) and \(l(w)=k\) is the length of \(w\). For \(x,y\in W\), we write \(x<_Ly\) if there is a reduced expression for \(y\) which ends with a reduced expression for \(x\). This partial ordering on \(W\) is called the (left) weak order. In [Trans. Am. Math. Soc. 308, No. 1, 1-37 (1988; Zbl 0659.05007)] \textit{A.~Björner} and \textit{M.~L.~Wachs} proved that under the weak order every quotient of a Coxeter group is a meet semi-lattice, and in the finite case is a lattice. In this paper, the author examines the case of an affine Weyl group \(W\) with corresponding finite Weyl group \(W_0\). In particular, the author shows that the quotient of \(W\) by \(W_0\) is a lattice and that up to isomorphism this is the only quotient of \(W\) which is a lattice. The author also determines that the question of which pairs of elements of \(W\) have upper bounds can be reduced to the analogous question within a particular finite subposet.
reviewed by: Chen Chengdong
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 05E15
Mathematics Subject Classification ID: 06A07
zbMATH DE Number: 1423854
zbMATH Keywords: affine Weyl groups
zbMATH Keywords: lattices
zbMATH Keywords: upper bounds
zbMATH Keywords: finitely generated Coxeter groups
zbMATH Keywords: reduced expressions
zbMATH Keywords: weak orders
MaRDI profile type: Publication
full work available at URL: a:1006352408688
OpenAlex ID: W19382891
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description: scientific article; zbMATH DE number 1423854