Abelianizing the real permutation action via blowups

2003/05/31 by Eva Maria Feichtner, Dmitry N. Kozlov
Mathematics · #math.AG #math.CO

paper · pdf

published as Int. Math. Res. Not. 2003, no. 32 (2003), 1755-1784 · 23 pages, 6 figures

arxiv created 2003/08/03 · arxiv updated 2009/11/30

Abstract

We present an abelianization of the permutation action of the symmetric group S_n on Rⁿ 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 Z₂. 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.

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