Abelianizing the real permutation action via blowups
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2003
Publication Type
Journal Article
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yes
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Abstract
We present an abelianization of the permutation action of the symmetric group Sn 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 Sn-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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2003 (32)
Pages / Article No.
1755 - 1784
Publisher
Hindawi
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Subject
ARRANGEMENTS; MODELS