Toward quantum Pieri rule for $F\ell_n$ via Seidel representation

TL;DR

This paper introduces a new proof of the Seidel operator on the quantum cohomology of flag varieties using a quantum-to-classical reduction, and reestablishes a quantum Pieri rule, also proposing a conjecture for quantum K-theory.

Contribution

It provides a novel proof of the Seidel operator and a quantum Pieri rule for flag varieties, along with a conjecture for quantum K-theory.

Findings

New proof of Seidel operator on quantum cohomology

Reproof of quantum Pieri rule for flag varieties

Conjecture on quantum Pieri rule in quantum K-theory

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 $QH^*(F\ell_n)$. Further, we reprove a quantum Pieri rule with respect to certain special Schubert class for $QH^*(F\ell_n)$. Finally, we propose a concrete conjecture on the corresponding quantum Pieri rule for the quantum $K$-theory of $F\ell_n$.