Order-Preserving Mappings Induced by Parabolic Decompositions on Coxeter Groups.

For a Coxeter system (W, S) and a subset J ⊆ S, fix u ∈ WJ and ξ ∈ WJ, let λ(−, u) : WJ → W and λ(ξ, −) : WJ → W be the mappings defined by λ(x, u) = xu and λ(ξ, y) = ξy, respectively. We show that λ(−, u) is left weak Bruhat order preserving for all u ∈ WJ, and that λ(ξ,−) is left weak Bruhat order-preserving if and only if it is a convex embedding, if and only if `(y) = `(ξyξ−1 ) for all y ∈ WJ. For the Coxeter groups of types A and B, we explicitly describe the ξ that satisfies the previous equivalent conditions in a purely combinatorial manner. [ABSTRACT FROM AUTHOR]