Abelianizing the real permutation action via blowups

Abstract

We present an abelianization of the permutation action of the symmetric group S_n on R^n in analogy to the Batyrev abelianization construction for finite group actions on complex manifolds. The abelianization is provided by a particular De Concini-Procesi wonderful model for the braid arrangement. In fact, we show a stronger result, namely that stabilizers of points in the arrangement model are isomorphic to direct products of Z_2. To prove that, we develop a combinatorial framework for explicitly describing the stabilizers in terms of automorphism groups of set diagrams over families of cubes. We observe that the natural nested set stratification on the arrangement model is not stabilizer distinguishing with respect to the S_n-action, that is, stabilizers of points are not in general isomorphic on open strata. Motivated by this structural deficiency, we furnish a new stratification of the De Concini-Procesi arrangement model that distinguishes stabilizers.