The Cohomology of Real Parabolic Arrangements

Abstract

Parabolic arrangements, obtained by deleting faces of the Coxeter complex of a reflection group, arise naturally in the study of real subspace arrangements. By bridging geometric methods for real arrangements (as in work of Baryshnikov, Dobrinskaya, and Turchin) with the combinatorial–algebraic viewpoint of k-parabolic arrangements (Barceló, Severs, and White), we derive explicit descriptions of the cohomology ring of complements of parabolic arrangements. Our main results introduce a chain complex generated by parabolic cosets whose homology computes the cohomology of the complement, together with an intersection-based cup product formula. This framework recovers and extends known results (including k-equal, and type B and D analogues) and applies to broader families of real subspace arrangements by interpreting linear conditions as unions of parabolic cones.