Freeness of Hyperplane Arrangements between Boolean Arrangements and Weyl Arrangements of Type Bℓ B_{\ell}

Every subarrangement of Weyl arrangements of type $B_{\ell}$ is represented by a signed graph. Edelman and Reiner characterized freeness of subarrangements between type $A_{\ell-1}$ and type $B_{\ell}$ in terms of graphs. Recently, Suyama and the authors characterized freeness for subarrangements containing Boolean arrangements satisfying a certain condition. This article is a sequel to the previous work. Namely, we give a complete characterization for freeness of arrangements between Boolean arrangements and Weyl arrangements of type $B_{\ell}$ in terms of graphs.Comment: 15 page

MAT-free graphic arrangements and a characterization of strongly chordal graphs by edge-labeling

Ideal subarrangements of a Weyl arrangement are proved to be free by the multiple addition theorem (MAT) due to Abe-Barakat-Cuntz-Hoge-Terao (2016). They form a significant class among Weyl subarrangements that are known to be free so far. The concept of MAT-free arrangements was introduced recently by Cuntz-M{\"u}cksch (2020) to capture a core of the MAT, which enlarges the ideal subarrangements from the perspective of freeness. The aim of this paper is to give a precise characterization of the MAT-freeness in the case of type $A$ Weyl subarrangements (or graphic arrangements). It is known that the ideal and free graphic arrangements correspond to the unit interval and chordal graphs respectively. We prove that a graphic arrangement is MAT-free if and only if the underlying graph is strongly chordal. In particular, it affirmatively answers a question of Cuntz-M{\"u}cksch that MAT-freeness is closed under taking localization in the case of graphic arrangements.Comment: 25 page

Worpitzky-compatible subarrangements of braid arrangements and cocomparability graphs

The class of Worpitzky-compatible subarrangements of a Weyl arrangement together with an associated Eulerian polynomial was recently introduced by Ashraf, Yoshinaga and the first author, which brings the characteristic and Ehrhart quasi-polynomials into one formula. The subarrangements of the braid arrangement, the Weyl arrangement of type $A$, are known as the graphic arrangements. We prove that the Worpitzky-compatible graphic arrangements are characterized by cocomparability graphs. Our main result yields new formulas for the chromatic and graphic Eulerian polynomials of cocomparability graphs.Comment: 11 pages, comments are welcome