The intermediate orders of a Coxeter group

Abstract

We define a class of partial orders on a Coxeter group that lie between the left weak order and the Bruhat order. We prove that these posets are graded by the length function and that the projections on the right parabolic quotients are always order preserving. We also introduce the notion of k-Bruhat graph, k-absolute length and k-absolute order, proposing some related conjectures and problems.