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

Abstract

Consider the natural torus action on a partial flag manifold $Fl$. Let $\hat{I}{\text{c◯}}_I\subset Fl$ be an open Schubert variety, and let $c^{sm}(\hat{I}{\text{c◯}}_I)\in H_T^{\ast }(Fl)$ be its torus equivariant Chern-Schwartz-MacPherson class. We show a set of interpolation properties that uniquely determine $c^{sm}(\hat{I}{\text{c◯}}_I)$, as well as a formula, of `localization type', for $c^{sm}(\hat{I}{\text{c◯}}_I)$. In fact, we proved similar results for a class $\hat{I}{º}_I\in H_T^{\ast }(Fl)$ --- in the context of quantum group actions on the equivariant cohomology groups of partial flag varieties. In this note we show that $c^{SM}(\hat{I}{\text{c◯}}_I)=\hat{I}{º}_I$.

9 pages, submitted to IMPANGA15, dedicated to 60th birthday of P. Pragacz, v2: new section added