# Dimers, crystals and quantum Kostka numbers

## TL;DR

This paper connects honeycomb dimer configurations on cylinders with quantum Kostka numbers, revealing new sum rules for Gromov-Witten invariants in the context of Grassmannian cohomology.

## Contribution

It establishes a novel relationship between dimer models, crystal graphs, and quantum cohomology, providing new computational tools.

## Key findings

- Dimer configurations correspond to vertices in crystal graphs.
- Quantum Kostka numbers are derived from dimer counts.
- Sum rules for Gromov-Witten invariants are obtained.

## Abstract

We relate the counting of honeycomb dimer configurations on the cylinder to the counting of certain vertices in Kirillov-Reshetikhin crystal graphs. We show that these dimer configurations yield the quantum Kostka numbers of the small quantum cohomology ring of the Grassmannian, i.e. the expansion coefficients when multiplying a Schubert class repeatedly with different Chern classes. This allows one to derive sum rules for Gromov-Witten invariants.