Toward quantum Pieri rule for $F{\mathrm{\ell }}_n$ via Seidel representation

By using a ``quantum-to-classical" reduction formula on the Gromov-Witten invariants of flag vaireities $F{\mathrm{\ell }}_n$, we provide a new proof of the Seidel operator on the quantum cohomology ring $Q{H}^{\ast }(F{\mathrm{\ell }}_n)$. Further, we reprove a quantum Pieri rule with respect to certain special Schubert class for $Q{H}^{\ast }(F{\mathrm{\ell }}_n)$. Finally, we propose a concrete conjecture on the corresponding quantum Pieri rule for the quantum $K$-theory of $F{\mathrm{\ell }}_n$.