Journal of Algebraic Combinatorics

, Volume 47, Issue 2, pp 213–231 | Cite as

Chern class of Schubert cells in the flag manifold and related algebras

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Abstract

We discuss a relationship between Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds, the Fomin–Kirillov algebra, and the generalized nil-Hecke algebra. We show that the nonnegativity conjecture in the Fomin–Kirillov algebra implies the nonnegativity of the Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds for type A. Motivated by this connection, we also prove that the (equivariant) Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds are certain summations of the structure constants of the equivariant cohomology of Bott–Samelson varieties. We also discuss refined positivity conjectures of the Chern–Schwartz–MacPherson classes for Schubert cells motivated by the nonnegativity conjecture in the Fomin–Kirillov algebra.

Keywords

Bott–Samelson variety Chern–Schwartz–MacPherson class Flag variety Fomin–Kirillov algebra Generalized nil-Hecke algebra 
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Notes

Acknowledgements

I thank Leonardo Mihalcea for helpful discussions. Funding was provided by KIAS.

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Korea Institute for Advanced StudySeoulRepublic of Korea

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