The freeness of Shi-Catalan arrangements

Details The freeness of Shi-Catalan arrangements The freeness of Shi-Catalan arrangements

2010-12-29

arxiv.org/abs/1012.5884v1

European Journal of Combinatorics, Vol. 32, no. 8, 1191-1198, 2011

Mathematics

Combinatorics

12 pages

Scientific paper

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.

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