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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)].
The following theorem is the main result of this article:
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.

MSC:
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
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