Equivariant Chern–Schwartz–MacPherson classes in partial flag varieties: interpolation and formulae

Abstract

Consider the natural torus action on a partial flag manifold \( \mathcal F}_\lambda \). Let \( \Omega_I\subset \mathcal F}_\lambda \) be an open Schubert variety, and let \( c^{sm}(\Omega_I)\in H_T^*(\mathcal F}_\lambda) \) be its torus equivariant Chern–Schwartz–MacPherson class. We show a set of interpolation properties that uniquely determine $c^{sm}({\Omega }_I)$, as well as a formula, of 'localization type', for $c^{sm}({\Omega }_I)$. In fact, we proved similar results for a class \( \kappa_I\in H_T^*(\mathcal F}_\lambda) \) – in the context of quantum group actions on the equivariant cohomology groups of partial flag varieties. In this note we show that $c^{sm}({\Omega }_I)={\kappa }_I$.