Schubert calculus and the Hopf algebra structures of exceptional Lie groups

TL;DR

This paper develops a unified approach to understanding the cohomology rings of exceptional Lie groups as Hopf algebras over the Steenrod algebra, leveraging Schubert calculus and Weyl invariants.

Contribution

It introduces a new unified method to analyze the Hopf algebra structure of cohomology rings of exceptional Lie groups using Schubert calculus and Weyl invariants.

Findings

Unified description of H*(G;F_{p}) as a Hopf algebra over A_{p}

Application to integral cohomology of all exceptional Lie groups

Determination of near-Hopf ring structures

Abstract

Let G be an exceptional Lie group with a maximal torus T. Based on common properties in the Schubert presentation of the cohomology ring H*(G/T;F_{p}) DZ1, and concrete expressions of generalized Weyl invariants for G over F_{p}, we obtain a unified approach to the structure of H*(G;F_{p}) as a Hopf algebra over the Steenrod algebra A_{p}. The results has been applied in Du2 to determine the near--Hopf ring structure on the integral cohomology of all exceptional Lie groups.