Poset pinball, GKM-compatible subspaces, and Hessenberg varieties
Poset pinball, GKM-compatible subspaces, and Hessenberg varieties
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Poset pinball、GKM 兼容子空间和 Hessenberg 簇
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发表时间：
2010
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通讯作者：
Julianna Tymoczko
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作者：
M. Harada;Julianna Tymoczko
本文有三个主要目标。首先，我们建立了一个一般的框架，以解决问题的等变上同调的某些子空间的GKM空间的模基的构造。为此，我们引入的概念，GKM相容子空间的环境GKM空间。我们还讨论了偏序上三角，一个关键的组合概念，在GKM理论和更普遍的本地化理论在等变上同调。鉴于对其他应用程序，我们目前的部分，我们的设置在一个一般的代数和组合的框架。其次，我们的中心问题的动机，建立模块基地，我们介绍了一个组合游戏，我们称之为偏序集弹球，并说明了几个例子。最后，作为第一个应用，我们应用GKM相容子空间和偏序集弹球的观点来构造所有经典李型的Peterson簇和李型$A$的次正则Springer簇的S^1 $-等变上同调的显式和计算方便的模基。此外，在Springer情形下，我们利用我们的模基将次正则Springer簇的普通上同调的经典Springer表示提升到李型$A$中的S^1 $-等变上同调。
This paper has three main goals. First, we set up a general framework to address the problem of constructing module bases for the equivariant cohomology of certain subspaces of GKM spaces. To this end we introduce the notion of a GKM-compatible subspace of an ambient GKM space. We also discuss poset-upper-triangularity, a key combinatorial notion in both GKM theory and more generally in localization theory in equivariant cohomology. With a view toward other applications, we present parts of our setup in a general algebraic and combinatorial framework. Second, motivated by our central problem of building module bases, we introduce a combinatorial game which we dub poset pinball and illustrate with several examples. Finally, as first applications, we apply the perspective of GKM-compatible subspaces and poset pinball to construct explicit and computationally convenient module bases for the $S^1$-equivariant cohomology of all Peterson varieties of classical Lie type, and subregular Springer varieties of Lie type $A$. In addition, in the Springer case we use our module basis to lift the classical Springer representation on the ordinary cohomology of subregular Springer varieties to $S^1$-equivariant cohomology in Lie type $A$.