Computing Chow rings of twisted flag varieties for subgroups of an algebraic group

TL;DR

This paper develops methods to compute the Chow rings of twisted flag varieties associated with algebraic groups over various fields, advancing understanding of their algebraic and geometric structures.

Contribution

It introduces new techniques for calculating Chow rings of twisted flag varieties for different algebraic groups over extension fields.

Findings

Explicit formulas for Chow rings of certain twisted flag varieties

New computational methods for algebraic cycles in twisted settings

Enhanced understanding of the algebraic structure of flag varieties

Abstract

Let $G_k$ be a split algebraic group over $k$. Given a non-trivial $G_k$-torsor $G$, we consider the (twisted) flag variety $F=G/B_k$ for the Borel subgroup $B_k$ containing in $G_k$. The purpose of this paper is the study how to compute the Chow rings $CH^*(F)$ for various $F$ over some extension fields $K$ of $k$.