Schubert calculus and the Hopf algebra structures of exceptional Lie groups

Comodule Structures, Equivariant Hopf Structures, and Generalized Schubert Polynomials

In this article, the comodule structure of Chow rings of Flag manifolds $\operatorname{CH}(G/B)$ is described by Schubert cells. Its equivariant version gives rise to a Hopf structure of the equivariant cohomology of flag manifolds $H^*_B(G/B)$. We get two identities of generalized Schubert polynomials as explanations of the geometric facts.

Schubert presentation of the integral cohomology ring of the flag manifolds G/T

Let G be a compact connected Lie group with a maximal torus T\subsetG. In the context of Schubert calculus we obtain a canonical presentation for the integral cohomology ring H^{\ast}(G/T) of the complete flag manifold G/T. The result have been applied in [15] to construct the integral cohomology ring H^{\ast}(G) in terms of Schubert classes on G/T, and in [16] to determine the structure of the modp cohomology H^{\ast}(G;F_{p}) as a Hopf algebra over the Steenrod algebra.

The near-Hopf ring structure on the integral cohomology of 1-connected Lie groups

Let G be an exceptional Lie group with a maximal torus T. Based on Schubert calculus on the flag manifold G/T we have described the integral cohomology ring H*(G) by explicitely constructed generators in [DZ2], and determined the structure of H*(G;F_{p}) as a Hopf algebra over the Steenrod algebra in[DZ3]. In this sequel to [DZ2], [DZ3] we obtain the near--Hopf ring structure on H*(G).

Homology rigidity of Grassmannians

Applying the theory of Gröbner basis to the Schubert presentation of the cohomology of Grassmannians, we extend the homology rigidity results known for the classical Grassmannians to the exceptional cases.

The Chow rings of the algebraic groups E_6, E_7, and E_8

We determine the Chow rings of the complex algebraic groups of the exceptional type E_6, E_7, and E_8, giving the explicit generators represented by the pull-back images of Schubert varieties of the corresponding flag varieties. This is a continuation of the work of R. Marlin on the computation of the Chow rings of SO_n, Spin_n, G_2, and F_4. Our method is based on Schubert calculus of the corresponding flag varieties, which has its own interest.

格拉斯曼型舒伯特簇的拟对称模拟

A quasisymmetric analog of Grassmannian Schubert varieties

本文证明格拉斯曼流形中环面理查森簇的上同调环为拟对称函数环的有限截断，构造仿射铺砌，并利用直接极限得到具有弱H-群结构的ind-簇，其上同调为拟对称函数的Hopf代数，同时推出碎片多面体的f-向量对数凹性。