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

By Paolo Aluffi, Leonardo Constantin Mihalcea. Compositio Mathematica, 2015

46 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/17c980a8b6b23dfeb5c85859d0ddf442b2607f86

How researchers see this paper

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In Discrete Mathematics and Combinatorics

1. Leonardo C.Mihalcea
2. Joerg Schuermann
3. C. Joanna Su
4. Paolo Aluffi

In Geometry and Topology

1. Leonardo C.Mihalcea
2. Joerg Schuermann
3. C. Joanna Su
4. Paolo Aluffi
5. Eric Katz

In Mathematical Physics

1. Leonardo C.Mihalcea
2. Joerg Schuermann
3. C. Joanna Su
4. Paolo Aluffi
5. Eric Katz

Citations in the corpus

• Shadows of characteristic cycles, Verma modules, and positivity of Chern–Schwartz–MacPherson classes of Schubert cells (2017, methodology): “specialization Li := (Li)~7→1 is an operator which was shown in [5] to generate recursively the CSM classes.”