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 (⁎).