The Parabolic Map

In this paper we study certain aspects of Bruhat order on Coxeter groups. LetWbe a Coxeter group, and let denote the set of parabolic subgroups ofW. We construct a mapm : W × → Wand examine its first properties. We prove the following characterization:wis the longest element of a parabolic subgroup ofWif and only if for anyv W, there exists a unique maximal element (with respect to Bruhat order) less than or equal to bothvandw.