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

TL;DR

This paper provides a complete graph-theoretic characterization of the freeness of hyperplane arrangements situated between Boolean and Weyl arrangements of type B, extending previous work on related arrangements.

Contribution

It offers a comprehensive characterization of freeness for arrangements between Boolean and Weyl type B arrangements using graph theory, building on prior partial results.

Findings

Freeness characterized by signed graphs.

Complete classification for arrangements between Boolean and Weyl B arrangements.

Extends previous partial characterizations.

Abstract

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.