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 (?).