  We discuss a relationship between Chern-Schwartz-MacPherson classes for
Schubert cells in flag manifolds, Fomin-Kirillov algebra, and the generalized
nil-Hecke algebra. We show that nonnegativity conjecture in 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 the Bott-Samelson varieties. We also
discuss the refined positivity conjectures of the Chern-Schwartz-MacPherson
classes for Schubert cells motivated by the nonnegativity conjecture in
Fomin-Kirillov algebra.
