The equivariant Orlik–Solomon algebra

Given a real hyperplane arrangement , the complement of the complexification of admits an action of the group by complex conjugation. We define the equivariant Orlik–Solomon algebra of to be the -equivariant cohomology ring of with coefficients in the field . We give a combinatorial presentation of this ring, and interpret it as a deformation of the ordinary Orlik–Solomon algebra into the Varchenko–Gelfand ring of locally constant -valued functions on the complement of in . We also show that the -equivariant homotopy type of is determined by the oriented matroid of . As an application, we give two examples of pairs of arrangements and such that and have the same nonequivariant homotopy type, but are distinguished by the equivariant Orlik–Solomon algebra.