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 conjectures to the torus equivariant setting.

This version includes the equivariant case

References in corpus (1)

• A Schubert calculus recurrence from the noncomplex W-action on G/B

Cited by in corpus (13)

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• Elliptic classes of Schubert varieties via Bott-Samelson resolution
• Left Demazure-Lusztig operators on equivariant (quantum) cohomology and K theory
• Positivity of Segre-MacPherson classes
• The Chern-Schwartz-MacPherson class of an embeddable scheme
• From motivic Chern classes of Schubert cells to their Hirzebruch and CSM classes
• Structure Constants in equivariant oriented cohomology of flag varieties
• Stable bases of the Springer resolution and representation theory
• Geometric properties of the Kazhdan-Lusztig Schubert basis
• Chern Classes of Open Projected Richardson Varieties and of Affine Schubert Cells
• On equivariant oriented cohomology of Bott-Samelson varieties