Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds

Abstract

We obtain an algorithm computing the Chern–Schwartz–MacPherson (CSM) classes of Schubert cells in a generalized flag manifold$G/B$. In analogy to how the ordinary divided difference operators act on Schubert classes, each CSM class of a Schubert class is obtained by applying certain Demazure–Lusztig-type operators to the CSM class of a cell of dimension one less. These operators define a representation of the Weyl group on the homology of$G/B$. By functoriality, we deduce algorithmic expressions for CSM classes of Schubert cells in any flag manifold$G/P$. We conjecture that the CSM classes of Schubert cells are an effective combination of (homology) Schubert classes, and prove that this is the case in several classes of examples. We also extend our results and conjecture to the torus equivariant setting.