Representation ring of Levi subgroups versus cohomology ring of flag varieties

TL;DR

This paper explores the relationship between the representation rings of Levi subgroups and the cohomology rings of flag varieties, generalizing a classical result linking Grassmannian cohomology and polynomial representations of GL(r).

Contribution

It proposes a generalization of the classical isomorphism between representation rings and cohomology rings from GL(r) and Grassmannians to Levi subgroups of arbitrary reductive groups and their flag varieties.

Findings

Established a framework for relating Levi subgroup representation rings to flag variety cohomology rings.

Extended classical results from GL(r) to more general reductive groups.

Provided explicit ring homomorphisms connecting these algebraic and geometric structures.

Abstract

Recall the classical result that the cup product structure constants for the singular cohomology with integral coefficients of the Grassmannian of r-planes coincide with the Littlewood-Richardson tensor product structure constants for GL(r). Specifically, the result asserts that there is an explicit ring homomorphism \phi: \Rep _{\poly}(GL(r)) to H^*(Gr(r, n)), where Gr(r, n) denotes the Grassmannian of r-planes in C^n and \Rep_{\poly} (GL(r)) denotes the polynomial representation ring of GL(r). This work seeks to achieve one possible generalization of this classical result for GL(r) and the Grassmannian Gr(r,n) to the Levi subgroups of any reductive group G and the corresponding flag varieties.