Three presentations of torus equivariant cohomology of flag manifolds
Shizuo Kaji

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 be a compact connected Lie group and be its maximal torus. The homogeneous space is called the (complete) flag manifold. One of the main goals of the {\em equivariant Schubert calculus} is to study the -equivariant cohomology with regard to the -action on by multiplication. There are three presentations known for ; (1) the free -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…
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Advanced Algebra and Geometry · Algebraic structures and combinatorial models
