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

Part of: Projective and enumerative geometry Special varieties Cycles and subschemes

Published online by Cambridge University Press:  14 November 2016

Paolo Aluffi  and
Leonardo C. Mihalcea
Paolo Aluffi
Affiliation:
Mathematics Department, Florida State University, Tallahassee, FL 32306, USA email aluffi@math.fsu.edu
Leonardo C. Mihalcea
Affiliation:
Department of Mathematics, Virginia Tech University, Blacksburg, VA 24061, USA email lmihalce@vt.edu
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Abstract

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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.

Keywords

Chern–Schwartz–MacPherson classhomogeneous spaceSchubert varietyDemazure–Lusztig operator

MSC classification

Primary: 14M15: Grassmannians, Schubert varieties, flag manifolds
Secondary: 14C17: Intersection theory, characteristic classes, intersection multiplicities 14N15: Classical problems, Schubert calculus
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 12 , December 2016 , pp. 2603 - 2625
Copyright
© The Authors 2016 

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