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The freeness of Shi-Catalan arrangements. (English) Zbl 1235.52035
Let $$W$$ be a finite Weyl group and $${\mathcal A}$$ be the corresponding Weyl arrangement. For two multiplicities $${\mathbf a}$$ and $${\mathbf b}$$, a deformation $${\mathcal A}^{[-{\mathbf a}, {\mathbf b}]}$$ of $${\mathcal A}$$ is an affine arrangement which is obtained by adding to each hyperplane $$H \in{\mathcal A}$$ several parallel translations of $$H$$ by the positive root (and its integer multiples) perpendicular to $$H$$. A deformation is $$W$$-equivariant if the number of parallel hyperplanes of each hyperplane $$H\in {\mathcal A}$$ depends only on the $$W$$-orbit of $$H$$.
The authors 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 [M. Yoshinaga, Invent. Math. 157, No. 2, 449–454 (2004; Zbl 1113.52039)].
Theorem. If $${\mathcal A}^{[-{\mathbf a}, {\mathbf b}]}$$ is a Shi-Catalan arrangement, then its coning $${\mathbf c}({\mathcal A}^{[-{\mathbf a}, {\mathbf b}]})$$ is free.
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