Based on a"quantum-to-classical"reduction formula of the Gromov-Witten invariants for the fiag variety Fℓn,we provide a new proof of the formula of a Seidel operator that acts on the quantum cohomology ring QH*(Fℓn).Furthermore,we reprove a quantum Pieri rule with respect to certain special Schubert class for QH*(Fℓn).We also propose a concrete conjecture on the corresponding quantum Pieri rule for the quantum K-theory of Fℓn.
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