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We discuss a relationship between Chern-Schwartz-MacPherson classes forSchubert cells in flag manifolds, Fomin-Kirillov algebra, and the generalizednil-Hecke algebra. We show that nonnegativity conjecture in Fomin-Kirillovalgebra implies the nonnegativity of the Chern-Schwartz-MacPherson classes forSchubert cells in flag manifolds for type A. Motivated by this connection, wealso prove that the (equivariant) Chern-Schwartz-MacPherson classes forSchubert cells in flag manifolds are certain summations of the structureconstants of the equivariant cohomology of the Bott-Samelson varieties. We alsodiscuss the refined positivity conjectures of the Chern-Schwartz-MacPhersonclasses for Schubert cells motivated by the nonnegativity conjecture inFomin-Kirillov algebra.

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