Abelianizing the Real Permutation Action via Blowups

Our object of study is an abelianization of the Sn permutation action on R n that is provided by a particular De Concini-Procesi wonderful model for the braid arrangement. Our motivation comes from an analogous construction for finite group actions on complex manifolds, due to Batyrev [B1, B2], and subsequent study of Borisov & Gunnells [BG], where the connection of such abelianizations with De Concini-Procesi wonderful models for arrangement complements was first observed. Whereas previous studies were restricted to complex manifolds, here we study one of the most natural nontrivial actions of a finite group on a real differentiable manifold, namely the permutation action on Rn. The locus of non-trivial stabilizers in this case is provided by the braid arrangement An−1. We suggest to blow up intersections of subspaces in An−1, respectively proper transforms of those intersections, in the order of an arbitrary linear extension of the intersection lattice Πn, so as to exhaust all of the arrangement. That is the same as to take the De Concini-Procesi wonderful model of the arrangement complement with respect to the maximal building set, see [DP]. Not only do we obtain an abelianization of the real permutation action, we even show that stabilizers of points in the arrangement model are isomorphic to direct products of Z2. To this end, we develop a combinatorial framework for explicitly describing the stabilizers in terms of automorphism groups of set diagrams over families of cubes. Moreover, we observe that the natural nested set stratification on the arrangement model is not stabilizer distinguishing with respect to the Sn-action, i.e., 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. Arrangement models have been extensively studied over the last years. They were introduced by De Concini & Procesi in [DP], one of the motivations being to provide rational models for cohomology algebras of arrangement complements. In [FK] the De Concini-Procesi model construction was put in a very general combinatorial context, showing that the notions of building sets and nested sets, coined already by Fulton & MacPherson in [FM], along with the notion of a blowup, have canonical combinatorial counterparts in the theory of semilattices. It was also shown in [FK] that this combinatorial framework actually traces precisely the step-by-step change in the incidence structure of strata during the De Concini-Procesi resolution process.