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

Abstract:

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.