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

Abstract

For an arrangement of linear subspaces in R n \mathbb {R}^n that is invariant under a finite subgroup of the general linear group G l n ( R ) Gl_n(\mathbb {R}) we develop a formula for the G G -module structure of the cohomology of the complement M A \mathcal {M}_{\mathcal {A}} . Our formula specializes to the well known Goresky-MacPherson theorem in case G = 1 G = 1 , but for G ≠ 1 G \neq 1 the formula shows that the G 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 M A / G \mathcal {M}_{\mathcal {A}}/G . Our motivating examples are arrangements in C n \mathbb {C}^n that are invariant under the action of S n S_n by permuting coordinates. A particular case is the “ k k -equal” arrangement, first studied by Björner, Lovász, and Yao motivated by questions in complexity theory. In these cases M A \mathcal {M}_{\mathcal {A}} and M A / S n \mathcal {M}_{\mathcal {A}}/S_n are spaces of ordered and unordered point configurations in C n \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 R d \mathbb {R}^d and provide explicit results for the “ k k -equal” and the “ k k -divisible” cases.