Comodule Structures, Equivariant Hopf Structures, and Generalized Schubert Polynomials

TL;DR

This paper explores the comodule and Hopf algebra structures of Chow rings and equivariant cohomology of flag manifolds, providing new identities for generalized Schubert polynomials based on geometric insights.

Contribution

It introduces a novel description of comodule structures and establishes Hopf structures in equivariant cohomology, along with new identities for generalized Schubert polynomials.

Findings

Comodule structure of Chow rings described via Schubert cells.

Equivariant cohomology of flag manifolds has a Hopf algebra structure.

Derived identities for generalized Schubert polynomials.

Abstract

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.