A quantum Murnaghan--Nakayama rule for the flag manifold

TL;DR

This paper develops a quantum Murnaghan--Nakayama rule for the flag manifold's quantum cohomology, linking quantum products with classical ones through detailed combinatorial analysis.

Contribution

It introduces a new rule for multiplying Schubert classes by tautological classes in quantum cohomology, extending classical combinatorial methods to the quantum setting.

Findings

Derived a formula for quantum Schur polynomial multiplication by hook partitions.

Connected quantum and classical products via quantum Bruhat order analysis.

Provided tools for computing quantum products in flag manifold cohomology.

Abstract

In this paper, we give a rule for the multiplication of a Schubert class by a tautological class in the (small) quantum cohomology ring of the flag manifold. As an intermediate step, we establish a formula for the multiplication of a Schubert class by a quantum Schur polynomial indexed by a hook partition. This entails a detailed analysis of chains and intervals in the quantum Bruhat order. This analysis allows us to use results of Leung--Li and of Postnikov to reduce quantum products by hook Schur polynomials to the (known) classical product.