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
Equivariant Chern–Schwartz–MacPherson classes in partial flag varieties: interpolation and formulae cover
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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 csm(ΩI)c^{sm}(\Omega_I), as well as a formula, of 'localization type', for csm(ΩI)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 csm(ΩI)=κIc^{sm}(\Omega_I)=\kappa_I.