Journal of Algebraic Combinatorics

, Volume 11, Issue 3, pp 227–240 | Cite as

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

  • G. Denham
Article

Abstract

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 \(\mathfrak{S}_n \)-module structure of the kernel of B(q) at certain roots of unity.

hyperplane arrangement Milnor fibre quantum bilinear form braid arrangement 

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Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • G. Denham
    • 1
  1. 1.Department of MathematicsUniversity of MichiganAnn Arbor

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