Rationally smooth Schubert varieties and inversion hyperplane arrangements

Abstract

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 (d1+1)⋯(dl+1) of the coexponents d1,…,dl 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.