Linearly Shellable Complexes

We introduce the class of linearly shellable pure simplicial complexes. The characterizing property is the existence of a labeling of their vertices such that all linear extensions of the Bruhat order on the set of facets are shelling orders. Coxeter complexes of weak intervals and lower Bruhat intervals of parabolic right quotients, as type-selected Coxeter complexes of lower Bruhat intervals of parabolic left quotients, are proved to be linearly shellable. We also introduce the notion of linear strong shellability.