Inversion arrangements and Bruhat intervals

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Let W be a finite reflection 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 (*).