We show that an element w of a finite Weyl group W is rationally smooth if and only if the hyperplane arrangement I associated to the inversion set of w is inductively free, and the product (d<inf>1</inf>+1)⋯(d<inf>l</inf>+1) of the coexponents d<inf>1</inf>, . . ., d<inf>l</inf> is equal to the size of the Bruhat interval [e,w], where e is the identity in W. As part of the proof, we describe exactly when a rationally smooth element in a finite Weyl group has a chain Billey-Postnikov decomposition. For finite Coxeter groups, we show that chain Billey-Postnikov decompositions are connected with certain modular coatoms of I.
Slofstra, W. (2015). Rationally smooth Schubert varieties and inversion hyperplane arrangements. Advances in Mathematics, 285, 709–736. https://doi.org/10.1016/j.aim.2015.07.034