Presenting the cohomology of a Schubert variety

TL;DR

This paper provides a new, uniform presentation of the cohomology rings of Schubert varieties within flag manifolds, extending classical results and refining them specifically for type A cases.

Contribution

It introduces a root-system uniform approach using the essential set of Coxeter group elements, generalizing Fulton's permutation-based definition.

Findings

Unified presentation for cohomology of Schubert varieties

New characterization of essential sets in Coxeter groups

Refinements specific to type A cases

Abstract

We extend the short presentation due to [Borel '53] of the cohomology ring of a generalized flag manifold to a relatively short presentation of the cohomology of any of its Schubert varieties. Our result is stated in a root-system uniform manner by introducing the essential set of a Coxeter group element, generalizing and giving a new characterization of [Fulton '92]'s definition for permutations. Further refinements are obtained in type A.