The freeness of Shi-Catalan arrangements

الأصل · EN

Let W be a finite Weyl group and be the corresponding Weyl arrangement. A deformation of is an affine arrangement which is obtained by adding to each hyperplane H∈ 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∈ 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.