Colored partitions and a Generalization of the Braid Arrangement

Abstract

We study the topology and combinatorics of an arrangement of hyperplanes in ${\bf C}^n$ that generalizes the classical braid arrangement. The arrangement plays in important role in the work of Schechtman and Varchenko on Lie algebra homology, where it appears in a generic fiber of a projection of the braid arrangement. The study of the intersection lattice of the arrangement leads to the definition of lattices of colored partitions. A detailed combinatorial analysis then provides algebro-geometric and topological properties of the complement of the arrangement. Using results on the character of $S_n$ on the cohomology of these arrangements we are able to deduce the rational cohomology of certain spaces of polynomials in the complement of the standard discriminant that have no root in the first $s$ integers.