Group actions on arrangements of linear subspaces and applications to configuration spaces

For an arrangement of linear subspaces in ${\mathbb R} ^n$ that is invariant under a finite subgroup of the general linear group $Gl_n({\mathbb R} )$ we develop a formula for the $G$-module structure of the cohomology of the complement ${\mathcal M} _{\mathcal A}$. Our formula specializes to the well known Goresky-MacPherson theorem in case $G = 1$, but for $G \neq 1$ the formula shows that the $G$-module structure of the complement is not a combinatorial invariant. As an application we are able to describe the free part of the cohomology of the quotient space ${\mathcal M} _{\mathcal A} /G$. Our motivating examples are arrangements in ${\mathbb C} ^n$ that are invariant under the action of $S_n$ by permuting coordinates. A particular case is the ``$k$-equal'' arrangement, first studied by Björner, Lovász, and Yao motivated by questions in complexity theory. In these cases ${\mathcal M} _{\mathcal A}$ and ${\mathcal M} _{\mathcal A} /S_n$ are spaces of ordered and unordered point configurations in ${\mathbb C} ^n$ many of whose properties are reduced by our formulas to combinatorial questions in partition lattices. More generally, we treat point configurations in ${\mathbb R} ^d$ and provide explicit results for the ``$k$-equal'' and the ``$k$-divisible'' cases.