The freeness of Shi-Catalan arrangements

TL;DR

This paper proves that certain symmetric deformations of Weyl arrangements are free under a specific Shi-Catalan condition and provides a chamber count formula, extending Yoshinaga's theorem.

Contribution

It introduces a new class of free arrangements derived from Weyl arrangements via Shi-Catalan deformations, generalizing previous conjectures.

Findings

Conings of $W$-equivariant deformations are free arrangements.

Provides a formula for the number of chambers in these arrangements.

Generalizes Yoshinaga's theorem on Weyl 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 $H\in\A$ 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 $H\in \A$ 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.