The freeness of Shi–Catalan arrangements

Abstract

Let W be a finite Weyl group and A be the corresponding Weyl arrangement. A deformation of A is an affine arrangement which is obtained by adding to each hyperplane H∈A several parallel translations of H by the positive root (and its integer multiples) perpendicular to H. We say that a deformation is W-equivariant if the number of parallel hyperplanes of each hyperplane H∈A depends only on the W-orbit of H. We prove that the conings of the W-equivariant deformations are free arrangements under a Shi–Catalan condition and give a formula for the number of chambers. This generalizes Yoshinaga’s theorem conjectured by Edelman–Reiner.