A result on the Bruhat order of a Coxeter group (Q752848)

Let \(W=(W,S)\) be a Coxeter group with S the set of its Coxeter generators. Let \(\leq\) be the Bruhat order on W. To each \(w\in W\), we associate two subsets of S: \({\mathcal L}(w)=\{s\in S|\) \(sw<w\}\) and \({\mathcal R}(w)=\{s\in S|\) \(ws<w\}\). The main result of this paper is that if x,y\(\in W\) and \(s\in S\) satisfy the condition \(s\not\in {\mathcal L}(y)\cup {\mathcal R}(x)\) then \(xy<xsy\). As an application, this result is used to investigate some properties of the Hecke algebra associated to W and also to verify a conjecture of L. K. Jones.