Schubert calculus on flag varieties and its application

(1)确定了标志流形E_8/T的整上同环的环结构，其中E_8表示秩8的紧单连通简单例外李群，T表示其最大环面。我们还利用除差算子确定了产生这个环的Schubert类。利用这一结果，我们可以确定对应的复代数群E_8的Chow环。(2)利用局部化技术和齐次空间环面等变上同环的GKM描述，显式地计算了标志变量G_2/B和复二次项Q_n的环面等变上同环，其中G_2表示G_2型的复李群，B为Borel子群。(3)推广了Kono-Kozima的结果，证明了无限辛群Sp的基环空间的Pontrjagin环。无穷正交群SO与Schur P-环同构。问)函数

(1) We determined the ring structure of the integral cohomology ring of the flag manifold E_8/T, where E_8 denotes the compact simply-connected simple exceptional Lie group of rank 8 and T its maximal torus. We also identified the Schubert classes which generate this ring by means of the divided difference operators. Using this result, we were able to determine the Chow ring of the corresponding complex algebraic group E_8.(2) Using the localization technique and the GKM description of the torus equivariant cohomology rings of homogeneous spaces, we computed the torus equivariant cohomology ring of the flag variety G_2/B and the complex quadric Q_n explicitly, where G_2 denotes the complex Lie group of type G_2 and B a Borel subgroup.(3) Extending the result due to Kono-Kozima, we showed that the Pontrjagin ring of the based loop space of the infinite symplectic group Sp(resp. infinite orthogonal group SO) is isomorphic to the ring of Schur P-(resp. Q) functions