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

• ### Richárd Rimányi

University of North Carolina at Chapel Hill, USA
• ### Alexander Varchenko

University of North Carolina at Chapel Hill, USA

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## 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 $c^{sm}(\Omega_I)=\kappa_I$.