The parabolic map

Abstract

In this paper we study certain aspects of Bruhat order on Coxeter groups. Let W be a Coxeter group, and let P denote the set of parabolic subgroups of W. We construct a map m: W × P → Wand examine its first properties. We prove the following characterization: w is the longest element of a parabolic subgroup of W if and only if for any v ∈ W, there exists a unique maximal element (with respect to Bruhat order) less than or equal to both v and w. © 1999 Academic Press.