An equivariant basis for the cohomology of Springer fibers

TL;DR

This paper develops a combinatorial basis for the equivariant cohomology of Springer fibers in type A, generalizing previous monomial bases and enabling explicit computations of cohomology classes.

Contribution

It introduces a new basis for the equivariant cohomology of Springer fibers, extending known monomial bases and providing a framework for explicit cohomology calculations.

Findings

Defined a basis for equivariant cohomology of Springer fibers.

Connected the basis to Schubert classes and their images.

Provided a combinatorial approach to study cohomology rings.

Abstract

Springer fibers are subvarieties of the flag variety that play an important role in combinatorics and geometric representation theory. In this paper, we analyze the equivariant cohomology of Springer fibers for $GL_n(\mathbb{C})$ using results of Kumar and Procesi that describe this equivariant cohomology as a quotient ring. We define a basis for the equivariant cohomology of a Springer fiber, generalizing a monomial basis of the ordinary cohomology defined by De Concini and Procesi and studied by Garsia and Procesi. Our construction yields a combinatorial framework with which to study the equivariant and ordinary cohomology rings of Springer fibers. As an application, we identify an explicit collection of (equivariant) Schubert classes whose images in the (equivariant) cohomology ring of a given Springer fiber form a basis.