Three presentations of torus equivariant cohomology of flag manifolds

TL;DR

This paper provides explicit algorithms to convert between three known presentations of the torus-equivariant cohomology of flag manifolds, enabling computation of structure constants in Schubert calculus.

Contribution

It introduces a method to explicitly convert elements among three presentations of equivariant cohomology, facilitating calculations in Schubert calculus.

Findings

Algorithms for converting between presentations are explicitly described.

Implementation of the algorithms in Maple software.

Enhanced ability to compute equivariant structure constants.

Abstract

Let $G$ be a compact connected Lie group and $T$ be its maximal torus. The homogeneous space $G/T$ is called the (complete) flag manifold. One of the main goals of the {\em equivariant Schubert calculus} is to study the $T$-equivariant cohomology $H^*_T(G/T)$ with regard to the $T$-action on $G/T$ by multiplication. There are three presentations known for $H^*_T(G/T)$; (1) the free $H^*(BT)$-module generated by the Schubert varieties (2) (with the rational coefficients) the {\em double coinvariant ring} of the Weyl group (3) the {\em GKM ring} associated to the Hasse graph of the Weyl group. Each presentation has both advantages and disadvantages. In this paper, we describe how to convert an element in one presentation to another by giving an explicit algorithm, which can then be used to compute the equivariant structure constants for the product of Schubert classes. The algorithm is implemented in Maple.