Rationally smooth Schubert varieties and inversion hyperplane arrangements

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 $(d_1+1)\cdots (d_l+1)$ of the coexponents $d_1,\dots ,d_l$ 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 $I$.