Computations of the Comodule Structures of the Chow rings of Flag Varieties

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 ${\pi }^{\ast }:{\mathrm{CH}}^{\ast }(G/B;{\mathbb{F}}_p)⟶{\mathrm{CH}}^{\ast }(G;{\mathbb{F}}_p)$ 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 ${\pi }^{\ast }:{\mathrm{CH}}^{\ast }(G/B;{\mathbb{F}}_p)⟶{\mathrm{CH}}^{\ast }(G;{\mathbb{F}}_p)$, and some partial result of ${\pi }^{\ast }$ 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 ${\mu }^{\ast }:{\mathrm{CH}}^{\ast }(G/B;{\mathbb{F}}_p)⟶{\mathrm{CH}}^{\ast }(G\times G/B;{\mathbb{F}}_p)$.