Group actions on arrangements of linear subspaces and applications to configuration spaces
Group actions on arrangements of linear subspaces and applications to configuration spaces
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关于线性子空间排列和配置空间应用的群体行动
DOI：
10.1090/s0002-9947-97-01565-1
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发表时间：
1997
影响因子：
1.3
通讯作者：
V. Welker
中科院分区：
文献类型：
--
作者：
S. Sundaram;V. Welker
对于在一般线性群Gin（IR）的有限子群下不变的RI中线性子空间的排列，我们给出了补M. A.在G = 1的情况下，我们的公式专门适用于著名的Goresky-MacPherson定理，但对于G $& 1，该公式表明补的G-模结构不是组合不变量。作为一个应用，我们能够描述商空间MA 4 IG的上同调的自由部分。我们的动机的例子是安排在Cn下的Sn的置换坐标的作用下是不变的。一个特殊的例子是“k-相等”的安排，首先由Bj 6 rner，Lovasz和Yao在复杂性理论的问题的启发下研究。在这些情况下，MA 4和MA/Sn是Cn中有序和无序点配置的空间，其中许多性质通过我们的公式简化为分格中的组合问题。更一般地说，我们治疗点配置Rd和提供明确的结果为“k-相等”和“k-可分”的情况下。
For an arrangement of linear subspaces in RI that is invariant under a finite subgroup of the general linear group Gin(IR) we develop a formula for the G-module structure of the cohomology of the complement M.A. Our formula specializes to the well known Goresky-MacPherson theorem in case G = 1, but for G $& 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 MA4IG. Our motivating examples are arrangements in Cn that are invariant under the action of Sn by permuting coordinates. A particular case is the "k-equal" arrangement, first studied by Bj6rner, Lovasz, and Yao motivated by questions in complexity theory. In these cases MA4 and MA/Sn are spaces of ordered and unordered point configurations in Cn many of whose properties are reduced by our formulas to combinatorial questions in partition lattices. More generally, we treat point configurations in Rd and provide explicit results for the "k-equal" and the "k-divisible" cases.