  We study multiplication of any Schubert polynomial $\mathfrak{S}_w$ by a
Schur polynomial $s_\lambda$ (the Schubert polynomial of a Grassmannian
permutation) and the expansion of this product in the ring of Schubert
polynomials. We derive explicit nonnegative combinatorial expressions for the
expansion coefficients for certain special partitions $\lambda$, including
hooks and the 2x2 box. We also prove combinatorially the existence of such
nonnegative expansion when the Young diagram of $\lambda$ is a hook plus a box
at the (2,2) corner. We achieve this by evaluating Schubert polynomials at the
Dunkl elements of the Fomin-Kirillov algebra and proving special cases of the
nonnegativity conjecture of Fomin and Kirillov.
  This approach works in the more general setup of the (small) quantum
cohomology ring of the complex flag manifold and the corresponding (3-point)
Gromov-Witten invariants. We provide an algebro-combinatorial proof of the
nonnegativity of the Gromov-Witten invariants in these cases, and present
combinatorial expressions for these coefficients.
