  Given a flag variety $Fl(n;r_1, \dots , r_\rho)$, there is natural ring
morphism from the symmetric polynomial ring in $r_1$ variables to the quantum
cohomology of the flag variety. In this paper, we show that for a large class
of partitions $\lambda$, the image of $s_\lambda$ under the ring homomorphism
is a Schubert class which is described by partitioning $\lambda$ into a quantum
hook (or $q$-hook) and a tuple of smaller partitions. We use this result to
show that the Pl\"ucker coordinate mirror of the flag variety describes quantum
cohomology relations. This gives new insight into the structure of this
superpotential, and the relation between superpotentials of flag varieties and
those of Grassmannians (where the superpotential was introduced by
Marsh--Rietsch).
