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

TL;DR

This paper introduces an algorithm to compute Chern-Schwartz-MacPherson classes of Schubert cells in flag manifolds, using Demazure-Lusztig operators, and explores their properties and conjectures in various settings.

Contribution

It develops a new algorithm based on Demazure-Lusztig operators for computing CSM classes of Schubert cells in flag manifolds and extends these results to equivariant cases.

Findings

Algorithm for CSM classes using Demazure-Lusztig operators

Representation of the Weyl group on homology via these operators

Conjecture and partial proof that CSM classes are effective combinations of Schubert classes

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 conjectures to the torus equivariant setting.