Equivariant cohomology and the Varchenko-Gelfand filtration
Equivariant cohomology and the Varchenko-Gelfand filtration
复制标题
等变上同调和 Varchenko-Gelfand 过滤
DOI：
10.1016/j.jalgebra.2016.10.010
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发表时间：
2011
期刊：
影响因子：
--
通讯作者：
Daniel Moseley
中科院分区：
文献类型：
--
作者：
Daniel Moseley
R3中n点的位形空间的上同调与对称群的正则表示同构，对称群的正则表示通过置换这些点来起作用。给出了这一事实的一个新的证明，证明了上同调环典型地同构于R1中n点的配置空间上的上同调上同调的Varchenko-Gelfand滤子的伴随分次.同时，我们还给出了R3配置空间的等变上同调环的表示.我们将我们的结果推广到任意实超平面排列(上述定理对应于辫子排列)以及定向拟阵的设置。
The cohomology of the configuration space of n points in R 3 is isomorphic to the regular representation of the symmetric group, which acts by permuting the points. We give a new proof of this fact by showing that the cohomology ring is canonically isomorphic to the associated graded of the Varchenko–Gelfand filtration on the cohomology of the configuration space of n points in R 1. Along the way, we give a presentation of the equivariant cohomology ring of the R 3 configuration space with respect to a circle acting on R 3 via rotation around a fixed line. We extend our results to the settings of arbitrary real hyperplane arrangements (the aforementioned theorems correspond to the braid arrangement) as well as oriented matroids.