On free deformations of the braid arrangement
The main result of this paper concerns the (central) hyperplane arrangements between the cones of the braid arrangement and the Shi arrangement in \(\mathbb{R}^{n+1}\). Each such arrangement corresponds to a simple directed graph \(([n],S)\) with vertex set \([n]=\{1,\ldots ,n\}\) and edge set \(S\subset \{(j,i):1\leq i<j\leq n\}\) in the following way. It consists of the hyperplanes \(x_i-x_j=0\) for \(1\leq i<j\leq n\), and \(x_i-x_j-x_0=0\) for \((j,i)\in S\), together with the hyperplane \(x_0=0\). The author shows that such an arrangement is free (in the sense of \textit{H. Terao} [J. Fac. Sci., Univ. Tokyo, Sect. IA 72, 293-312 (1980; Zbl 0509.14006)]) if and only if the graph \(([n],S)\) does not contain any induced subgraph of one of the following two types: \((\{i,j,k\},\{(k,j),(j,i)\})\) or \((\{i,j,k,l\},\{(j,i),(l,k)\})\). The underlying arrangement in \(\mathbb{R}^n\) (obtained by setting \(x_0=1\)) is inductively free. Among other results, the author proves that the cones of the extended Shi arrangements are free, thereby verifying part of a conjecture of \textit{P. H. Edelman} and \textit{V. Reiner} [Discrete Comput. Geom. 15, No. 3, 307-340 (1996; Zbl 0853.52013)].
scientific article; zbMATH DE number 1117435