The equivariant cohomology of weighted flag orbifolds

TL;DR

This paper develops the Schubert calculus framework for weighted partial flag orbifolds, including formulas for weighted Schubert classes and polynomials, extending classical results to this new setting.

Contribution

It introduces weighted Schubert classes, Chevalley's formula, and weighted double Schubert polynomials for orbifolds, expanding the algebraic tools available for these geometric objects.

Findings

Established Chevalley's formula for weighted Schubert classes

Defined weighted double Schubert polynomials and derived Chevalley--Monk's formula

Extended classical Schubert calculus results to weighted flag orbifolds

Abstract

We describe the torus-equivariant cohomology of weighted partial flag orbifolds ${\mathrm{w}}\Sigma$ of type $A$. We establish counterparts of several results known for the partial flag variety that collectively constitute what we refer to as ``Schubert Calculus on ${\mathrm{w}}\Sigma$''. For the weighed Schubert classes in ${\mathrm{w}}\Sigma$, we give the Chevalley's formula. In addition, we define the weighted analogue of double Schubert polynomials and give the corresponding Chevalley--Monk's formula.