An equivariant quantum Pieri rule for the Grassmannian on cylindric shapes

TL;DR

This paper introduces a new equivariant quantum Pieri rule for the Grassmannian using cylindric shapes, simplifying calculations by directly encoding addable boxes without complex auxiliary computations.

Contribution

It provides a direct equivariant generalization of Postnikov's quantum Pieri rule, avoiding the need for calculations in other Grassmannians or flag varieties.

Findings

Provides a Graham-positive rule for cylindric shapes

Simplifies calculations by encoding addable boxes directly

Complements existing work in quantum integrable systems

Abstract

The quantum cohomology ring of the Grassmannian is determined by the quantum Pieri rule for multiplying by Schubert classes indexed by row or column-shaped partitions. We provide a direct equivariant generalization of Postnikov's quantum Pieri rule for the Grassmannian in terms of cylindric shapes, complementing related work of Gorbounov and Korff in quantum integrable systems. The equivariant terms in our Graham-positive rule simply encode the positions of all possible addable boxes within one cylindric skew diagram. As such, unlike the earlier equivariant quantum Pieri rule of Huang and Li and known equivariant quantum Littlewood-Richardson rules, our formula does not require any calculations in a different Grassmannian or two-step flag variety.