Computations of the Comodule Structures of the Chow rings of Flag Varieties (2102.06672v1)
Abstract: Let $G$ be a connected reductive group, and $G/B$ be its flag variety. Let $\pi:G\to G/B$ be the natural projection. In this paper, we developed an algorithm to describe the map $\pi* :\operatorname{CH}*(G/B;\mathbb{F}_p)\longrightarrow \operatorname{CH}*(G;\mathbb{F}_p)$ in terms of Schubert cells. Taking advantage of the Pieri rule, we give an explicit formula for $A$-type, $C$-type, $G_2$, $F_4$ of the cohomology map $\pi* :\operatorname{CH}*(G/B;\mathbb{F}_p)\longrightarrow \operatorname{CH}*(G;\mathbb{F}_p)$, and some partial result of $\pi*$ is given for $E_6$ and $E_7$. Denote the group action map $\mu:G\times G/B\to G/B$, we also give an explicit formula for $A$-type, $C$-type, $G_2$, $F_4$ of the cohomology map $\mu*: \operatorname{CH}*(G/B;\mathbb{F}_p)\longrightarrow \operatorname{CH}*(G\times G/B;\mathbb{F}_p)$.