On Free Deformations of the Braid Arrangement

There has been considerable interest in the past in analyzing specific families of hyperplane arrangements from the perspective of freeness. Examples of such families have primarily included classes of subarrangements of Coxeter arrangements. The subarrangements of the braid arrangement An , the Weyl arrangement of type An−1, are known as the graphical arrangements. They correspond naturally to graphs on n vertices. It follows mainly from the work of Stanley [14] and is recorded in [5, §3] that free graphical arrangements correspond to chordal graphs. Certain classes of arrangements between the root systems An−1 and Bn were studied by Józefiak and Sagan [9]. These arrangements can also be related to graphs. Edelman and Reiner [5] gave a complete classification of the free arrangements in this case and showed that they correspond to threshold graphs. In a more recent work [6] these authors classified free arrangements which arise as discriminantal arrangements of two-dimensional zonotopes with integer side lengths. We will be concerned with deformations of An . The combinatorics of such arrangements was first studied in a systematic way by Stanley and collaborators [15]. They are the affine arrangements which have each of their hyperplanes parallel to one of the hyperplanes xi−x j = 0 of An . A central role in what follows will be played by the Shi arrangement of type An−1, introduced by J.-Y. Shi in [13]. It is the arrangement of affine hyperplanes in Rn of the form