Chern-Schwartz-MacPherson classes for Schubert cells in flag manifolds
Paolo Aluffi, Leonardo Constantin Mihalcea
We obtain an algorithm describing the Chern-Schwartz-MacPherson (CSM) classes of Schubert cells in generalized flag manifolds $G/B$. In analogy to how the ordinary divided difference operators act on Schubert classes, each CSM class of a Schubert class of a Schubert cell is obtained by applying certain Demazure-Lusztig type operators to the CSM class of a cell of dimension one less. 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.