Inversion arrangements and Bruhat intervals

Let W be a finite Coxeter group. For a given w ∈ W , the following assertion may or may not be satisfied: (∗) The principal Bruhat order ideal of w contains as many elements as there are regions in the inversion hyperplane arrangement of w. We present a type independent combinatorial criterion which characterises the elements w ∈ W that satisfy (∗). A couple of immediate consequences are derived: (1) The criterion only involves the order ideal of w as an abstract poset. In this sense, (∗) is a poset-theoretic property. (2) For W of type A, another characterisation of (∗), in terms of pattern avoidance, was previously given in collaboration with Linusson, Shareshian and Sjöstrand. We obtain a short and simple proof of that result. (3) If W is a Weyl group and the Schubert variety indexed by w ∈ W is rationally smooth, then w satisfies (∗).