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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Group actions on arrangements of
linear subspaces and applications to
configuration spaces

Authors: Sheila Sundaram and Volkmar Welker
Journal: Trans. Amer. Math. Soc. 349 (1997), 1389-1420
MSC (1991): Primary 05E25, 57N65.; Secondary 20C30, 55M35
MathSciNet review: 1340186
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Abstract: For an arrangement of linear subspaces in ${\mathbb R} ^n$ that is invariant under a finite subgroup of the general linear group $Gl_n({\mathbb R} )$ we develop a formula for the $G$-module structure of the cohomology of the complement ${\mathcal M} _{\mathcal A} $. Our formula specializes to the well known Goresky-MacPherson theorem in case $G = 1$, but for $G \neq 1$ the formula shows that the $G$-module structure of the complement is not a combinatorial invariant. As an application we are able to describe the free part of the cohomology of the quotient space ${\mathcal M} _{\mathcal A} /G$. Our motivating examples are arrangements in ${\mathbb C} ^n$ that are invariant under the action of $S_n$ by permuting coordinates. A particular case is the ``$k$-equal'' arrangement, first studied by Björner, Lovász, and Yao motivated by questions in complexity theory. In these cases ${\mathcal M} _{\mathcal A} $ and ${\mathcal M} _{\mathcal A} /S_n$ are spaces of ordered and unordered point configurations in ${\mathbb C} ^n$ many of whose properties are reduced by our formulas to combinatorial questions in partition lattices. More generally, we treat point configurations in ${\mathbb R} ^d$ and provide explicit results for the ``$k$-equal'' and the ``$k$-divisible'' cases.

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Additional Information

Sheila Sundaram
Affiliation: Department of Mathematics and Computer Science, University of Miami, Coral Gables, Florida 33124
Address at time of publication: Department of Mathematics, Wesleyan University, Middletown, Connecticut 06459

Volkmar Welker
Affiliation: Institute for Experimental Mathematics, Ellernstr. 29, 45326 Essen, Germany

Keywords: Subspace arrangements, group action, poset homology, configuration spaces, homotopy limits, symmetric functions
Received by editor(s): January 1, 1965
Additional Notes: The author acknowledges support by the DFG while he was visiting scholar at MIT
Article copyright: © Copyright 1997 American Mathematical Society