GROUP-ACTIONS ON ARRANGEMENTS OF LINEAR-SUBSPACES AND APPLICATIONS TOCONFIGURATION-SPACES

For an arrangement of linear subspaces in R-n that is invariant under a finite subgroup of the general linear group Gl(n)(R) we develop a fo rmula for the G-module structure of the cohomology of the complement M -A. Our formula specializes to the well known Goresky-MacPherson theor em in case G = 1, but for G not equal 1 the formula shows that the G-m odule structure of the complement is not a combinatorial invariant. As an application we are able to describe the free part of the cohomolog y of the quotient space M-A/G. Our motivating examples are arrangement s in C-n that are invariant under the action of S-n by permuting coord inates. A particular case is the ''k-equal'' arrangement, first studie d by Bjorner, Lovasz, and Yao motivated by questions in complexity the ory. In these cases M-A and M-A/S-n are spaces of ordered and unordere d point configurations in C-n many of whose properties are reduced by our formulas to combinatorial questions in partition lattices. More ge nerally, we treat point configurations in R-d and provide explicit res ults for the ''k-equal'' and the ''k-divisible'' cases.