A Giambelli formula for the $S^1$-equivariant cohomology of type A Peterson varieties
A Giambelli formula for the $S^1$-equivariant cohomology of type A Peterson varieties
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A 型 Peterson 簇的 $S^1$ 等变上同调的 Giambelli 公式
DOI：
10.2140/involve.2012.5.115
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发表时间：
2010
期刊：
影响因子：
--
通讯作者：
M. Harada
中科院分区：
文献类型：
--
作者：
Darius Bayegan;M. Harada
本文的主要结果是得到了$A$Peterson簇的$S^1$等变上同调环上的Peterson Schubert类的Giambeli公式。我们的结果依赖于Harada和Tymoczko导出的Peterson Schubert类等变结构常数的Monk公式。此外，我们证明了H.Naruse观察到的两个事实：第一，Peterson簇的$S^1$等变上同调的乘法结构中出现的一些常数是第二类Stirling数；第二，Peterson Schubert类在某种意义上满足稳定性性质，类似于经典的等变Schubert类在FLAG簇的$T^1等变上同调中的稳定性.
The main result of this note is a Giambelli formula for the Peterson Schubert classes in the $S^1$-equivariant cohomology ring of a type $A$ Peterson variety. Our results depend on the Monk formula for the equivariant structure constants for the Peterson Schubert classes derived by Harada and Tymoczko. In addition, we give proofs of two facts observed by H. Naruse: firstly, that some constants which appear in the multiplicative structure of the $S^1$-equivariant cohomology of Peterson varieties are Stirling numbers of the second kind, and secondly, that the Peterson Schubert classes satisfy a stability property in a sense analogous to the stability of the classical equivariant Schubert classes in the $T$-equivariant cohomology of the flag variety.