Quantum hooks and mirror symmetry for flag varieties

TL;DR

This paper explores the relationship between quantum cohomology, mirror symmetry, and flag varieties, revealing how certain Schubert classes correspond to partitioning partitions into quantum hooks, and connecting superpotentials across different varieties.

Contribution

It demonstrates that for a broad class of partitions, Schubert classes in flag varieties can be described via quantum hooks, linking quantum cohomology relations to mirror superpotentials.

Findings

Schubert classes correspond to partitioning into quantum hooks.

The Plücker coordinate mirror encodes quantum cohomology relations.

Connections established between flag and Grassmannian superpotentials.

Abstract

Given a flag variety $Fl(n;r_1, \dots , r_\rho)$, there is natural ring morphism from the symmetric polynomial ring in $r_1$ variables to the quantum cohomology of the flag variety. In this paper, we show that for a large class of partitions $\lambda$, the image of $s_\lambda$ under the ring homomorphism is a Schubert class which is described by partitioning $\lambda$ into a quantum hook (or $q$-hook) and a tuple of smaller partitions. We use this result to show that the Pl\"ucker coordinate mirror of the flag variety describes quantum cohomology relations. This gives new insight into the structure of this superpotential, and the relation between superpotentials of flag varieties and those of Grassmannians (where the superpotential was introduced by Marsh--Rietsch).