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

Chern class of Schubert cells in the flag manifold and related algebras 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Algebraic Combinatorics Springer Journals

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

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Publisher
Springer US
Copyright
Copyright © 2017 by Springer Science+Business Media, LLC
Subject
Mathematics; Combinatorics; Convex and Discrete Geometry; Order, Lattices, Ordered Algebraic Structures; Computer Science, general; Group Theory and Generalizations
ISSN
0925-9899
eISSN
1572-9192
D.O.I.
10.1007/s10801-017-0773-3
Publisher site
See Article on Publisher Site

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.

Journal

Journal of Algebraic CombinatoricsSpringer Journals

Published: Jun 23, 2017

References

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