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

TL;DR

This paper develops algorithms and explicit formulas for computing the comodule structures of Chow rings of flag varieties for various types, enhancing understanding of their algebraic and geometric properties.

Contribution

It introduces an algorithm to describe the pullback map on Chow rings of flag varieties and provides explicit formulas for classical types, with partial results for exceptional types.

Findings

Explicit formulas for $ ext{CH}^*(G/B)$ to $ ext{CH}^*(G)$ maps in types A, C, G2, F4.

Algorithmic description of the pullback map $ ext{CH}^*(G/B) o ext{CH}^*(G)$.

Partial results for exceptional groups $E_6$ and $E_7$.

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 $\mu^*: \operatorname{CH}^*(G/B;\mathbb{F}_p)\longrightarrow \operatorname{CH}^*(G\times G/B;\mathbb{F}_p)$.