The $T$-equivariant Integral Cohomology Ring of $E_6/T$

TL;DR

This paper computes the equivariant integral cohomology ring of the complex flag manifold of type E6 by applying a combinatorial version of the Leray-Hirsch theorem to homogeneous spaces.

Contribution

It introduces a combinatorial proof of the equivariant Leray-Hirsch theorem and explicitly describes the cohomology ring of E6 flag manifold as a polynomial quotient.

Findings

Explicit description of the equivariant cohomology ring of E6/T

Application of combinatorial Leray-Hirsch theorem to homogeneous spaces

Representation of the cohomology ring as a polynomial quotient

Abstract

We prove the equivariant Leray-Hirsch theorem combinatorially for sufficiently good torus equivariant fiber bundles consisting of homogeneous spaces of Lie groups. We apply this theorem to determining the equivariant integral cohomology ring of the flag manifold of type $E_6$ and express it explicitly as a quotient ring of a polynomial ring.