The freeness of Shi–Catalan arrangements
Copyright policy )The freeness of Shi–Catalan arrangements
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.
12 pages
- Hokkaido Bunkyo University Japan
- Hokkaido University Japan
- Kyoto University Japan
- Hokkeido University Japan
Computational Theory and Mathematics, hyperplane arrangements, Shi-Catalan arrangements, FOS: Mathematics, Mathematics - Combinatorics, Geometry and Topology, Combinatorics (math.CO), Arrangements of points, flats, hyperplanes (aspects of discrete geometry), free arrangements, Theoretical Computer Science, 32S22
Computational Theory and Mathematics, hyperplane arrangements, Shi-Catalan arrangements, FOS: Mathematics, Mathematics - Combinatorics, Geometry and Topology, Combinatorics (math.CO), Arrangements of points, flats, hyperplanes (aspects of discrete geometry), free arrangements, Theoretical Computer Science, 32S22
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