Hanlon and Stanley’s Conjecture and the Milnor Fibre of a Braid Arrangement

Let A be a real arrangement of hyperplanes. Let B = B(q) be Varchenko’s quantum bilinear form of A, introduced [15], specialized so that all hyperplanes have weight q . B(q) is nonsingular for all complex q except certain roots of unity. Here, we examine the kernel of B at roots of unity in relation to the topology of the hyperplane singularity. We use Varchenko’s work [16] to relate B(q) to a Salvetti complex for the Milnor fibration of A. This paper’s main result is specific to the arrangement of reflecting hyperplanes associated with the An−1 root system. We use a geometric property of the Milnor fibre to resolve a conjecture due to Hanlon and Stanley regarding the Sn-module structure of the kernel of B(q) at certain roots of unity.