A Weyl arrangement is the hyperplane arrangement defined by a root system. Saito proved that every Weyl arrangement is free. The Weyl subarrangements of type A ℓ are represented by simple graphs. Stanley gave a characterization of freeness for this type of arrangements in terms of thier graph. In addition, The Weyl subarrangements of type B ℓ can be represented by signed graphs. A characterization of freeness for them is not known. However, characterizations of freeness for a few restricted classes are known. For instance, Edelman and Reiner characterized the freeness of the arrangements between type A ℓ−1 and type B ℓ . In this paper, we give a characterization of the freeness and supersolvability of the Weyl subarrangements of type B ℓ under certain assumption.

The Ish arrangement was introduced by Armstrong to give a new interpretation of the $q,t$-Catalan numbers of Garsia and Haiman. Armstrong and Rhoades showed that there are some striking similarities between the Shi arrangement and the Ish arrangement and posed some problems. One of them is whether the Ish arrangement is a free arrangement or not. In this paper, we verify that the Ish arrangement is supersolvable and hence free. Moreover, we give a necessary and sufficient condition for the deleted Ish arrangement to be free.Comment: We added Section 3 (v2). We corrected the proof of Theorem 1.3 and several typos (v3

The braid arrangement is the Coxeter arrangement of the type Aℓ. The Shi arrangement is an affine arrangement of hyperplanes consisting of the hyperplanes of the braid arrangement and their parallel translations. In this paper, we give an explicit basis construction for the derivation module of the cone over the Shi arrangement. The essential ingredient of our recipe is the Bernoulli polynomials.

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