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

Abstract

Let {\cal A} be a real arrangement of hyperplanes. Let B e B(q) be Varchenko's quantum bilinear form of {\cal 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 {\cal A}. This paper's main result is specific to the arrangement of reflecting hyperplanes associated with the A_n − 1 root system. We use a geometric property of the Milnor fibre to resolve a conjecture due to Hanlon and Stanley regarding the {\Fraktur C}_n-module structure of the kernel of B(q) at certain roots of unity.