Poset弹球、最高形式和（n-2，2）Springer变种
Poset pinball, highest forms, and (n-2,2) Springer varieties
摘要：
我们研究型$A$幂零黑森贝格品种配备了一个自然的$S^1$ -行动使用Tymoczko，Harada Tymoczko，和Bayegan原田介绍的技术，特别强调一类特殊的幂零施普林格品种对应的分区$λ= (n-2,2)$为$n \geq 4$。首先，我们定义的邻接对矩阵对应于任何填充的杨图与$n$盒与字母表$\{1,2,\ldots,n\}$。使用邻接对矩阵，我们更明确，也延长了一些报表有关的最高形式的线性算子在以前的工作Tymoczko。其次，对于幂零算子$N$和Hessenberg函数$h$，我们构造了幂零Hessenberg簇$\Hess(N,h)$的$S^1$ -不动点与Young图$λ_N$的$(h,λ_N)$ -允许填充集之间的显式双射.第三，我们使用偏序集弹球，Harada和Tymoczko介绍的组合游戏，研究$A$型Springer簇$\mathcal{S}_{(n-2,2)}$的$S^1$ -等变上同调与$n\geq 4$的形状为$(n-2,2)$的Young图。具体来说，我们使用的尺寸对算法贝提可接受的弹球描述Bayegan和原田指定的$T$ -等变上同调的标志品种$\mathcal{F}\ell ags(\C^n)$映射到$H^*_{S^1}(\mathcal{S}_{(n-2,2)})$下的投影$H^*_T(\mathcal{F}\ell ags(\C^n)) \to H^*_{S^1}(\mathcal{S}_{(n-2,2)})$的模块基础的等变舒伯特类的一个子集。我们的弹球模的基础不是偏序上三角形，这是第一个具体的例子在文献中。我们的证明的一个结果是，存在一个简单而明确的基础，将我们的基础转换为一个偏序上三角模基$H^*_{S^1}(\mathcal{S}_{(n-2,2)})$的变化。最后，我们对未来的工作提出开放性问题。
英文摘要：
We study type $A$ nilpotent Hessenberg varieties equipped with a natural $S^1$-action using techniques introduced by Tymoczko, Harada-Tymoczko, and Bayegan-Harada, with a particular emphasis on a special class of nilpotent Springer varieties corresponding to the partition $λ= (n-2,2)$ for $n \geq 4$. First we define the adjacent-pair matrix corresponding to any filling of a Young diagram with $n$ boxes with the alphabet $\{1,2,\ldots,n\}$. Using the adjacent-pair matrix we make more explicit and also extend some statements concerning highest forms of linear operators in previous work of Tymoczko. Second, for a nilpotent operator $N$ and Hessenberg function $h$, we construct an explicit bijection between the $S^1$-fixed points of the nilpotent Hessenberg variety $\Hess(N,h)$ and the set of $(h,λ_N)$-permissible fillings of the Young diagram $λ_N$. Third, we use poset pinball, the combinatorial game introduced by Harada and Tymoczko, to study the $S^1$-equivariant cohomology of type $A$ Springer varieties $\mathcal{S}_{(n-2,2)}$ associated to Young diagrams of shape $(n-2,2)$ for $n\geq 4$. Specifically, we use the dimension pair algorithm for Betti-acceptable pinball described by Bayegan and Harada to specify a subset of the equivariant Schubert classes in the $T$-equivariant cohomology of the flag variety $\mathcal{F}\ell ags(\C^n)$ which maps to a module basis of $H^*_{S^1}(\mathcal{S}_{(n-2,2)})$ under the projection $H^*_T(\mathcal{F}\ell ags(\C^n)) \to H^*_{S^1}(\mathcal{S}_{(n-2,2)})$. Our pinball module basis is not poset-upper-triangular; this is the first concrete such example in the literature. A consequence of our proof is that there exists a simple and explicit change of basis which transforms our basis to a poset-upper-triangular module basis for $H^*_{S^1}(\mathcal{S}_{(n-2,2)})$. We close with open questions for future work.
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