A positive Monk formula in the S1‐equivariant cohomology of type A Peterson varieties
A positive Monk formula in the S1‐equivariant cohomology of type A Peterson varieties
复制标题
A 型 Peterson 簇 S1 等变上同调中的正 Monk 公式
DOI：
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
Julianna Tymoczko
中科院分区：
文献类型：
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作者：
M. Harada;Julianna Tymoczko
Peterson变种是Hessenberg变种的一种特殊类型，已被Peterson、Kostant和Rietsch等人广泛研究，并与FLAG变种的量子上同调有关。在这篇手稿中，我们发展了一个推广的Schubert演算，特别是一个正的−-Monk公式，证明了An-Peterson簇Y的普通上同调和Borel-等变上同调关于由极大环面在旗簇上的标准作用产生的自然的S1-作用。据我们所知，这是第一个超越Kac-Moody旗帜变种G/P的正Schubert演算的例子。首先，我们确定了HS1*(Y)的一个计算上方便的基，我们称之为Peterson Schubert类的基。其次，我们得到了上同调度为2的Peterson Schubert类与任意Peterson Schubert类的乘积的一个明显正的积分Chvalley-Monk公式。Hs1*(Y)和H*(Y)都是2次生成的。最后，利用我们的Chvalley-Monk公式，我们给出了−-1Peterson簇的S1-等变上同调环Hs1*(Y)和普通上同调环H*(Y)的显式刻画。我们的方法既直接来自GKM(Goresky-Kottwitz-MacPherson)理论，也受到经典Schubert微积分的启发。我们讨论了几个有待解决的问题和未来工作的方向。
Peterson varieties are a special class of Hessenberg varieties that have been extensively studied, for example, by Peterson, Kostant, and Rietsch, in connection with the quantum cohomology of the flag variety. In this manuscript, we develop a generalized Schubert calculus, and in particular a positive Chevalley–Monk formula, for the ordinary and Borel‐equivariant cohomology of the Peterson variety Y in type An−1, with respect to a natural S1‐action arising from the standard action of the maximal torus on flag varieties. As far as we know, this is the first example of positive Schubert calculus beyond the realm of Kac–Moody flag varieties G/P. Our main results are as follows. First, we identify a computationally convenient basis of HS1* (Y), which we call the basis of Peterson Schubert classes. Second, we derive a manifestly positive, integral Chevalley–Monk formula for the product of a cohomology‐degree‐2 Peterson Schubert class with an arbitrary Peterson Schubert class. Both HS1* (Y) and H*(Y) are generated in degree 2. Finally, by using our Chevalley–Monk formula we give explicit descriptions (via generators and relations) of both the S1‐equivariant cohomology ring HS1* (Y) and the ordinary cohomology ring H*(Y) of the type An−1 Peterson variety. Our methods are both directly from and inspired by those of the GKM (Goresky–Kottwitz–MacPherson) theory and classical Schubert calculus. We discuss several open questions and directions for future work.