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:(⁎) incipal Bruhat order ideal of w contains as many elements as there are regions in the inversion hyperplane arrangement of w. esent a type independent combinatorial criterion which characterises the elements w ∈ W that satisfy (⁎). A couple of immediate consequences are derived:(1) iterion only involves the order ideal of w as an abstract poset. In this sense, (⁎) is a poset-theoretic property. 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. s a Weyl group and the Schubert variety indexed by w ∈ W is rationally smooth, then w satisfies (⁎).