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

Abstract

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

14 pages. This paper was written in Chinese, published at SCIENTIA SINICA Mathematica, 2024, 54(12): 2009-2022. This is an English translation for those who might be interested