This talk will explain a combinatorial connection between exact matchings of the honeycomb (dimers), certain directed coloured graphs appearing in the representation theory of quantum groups (crystals) and the multiplication of a Schubert class by a Chern class of the tautological or quotient bundle in the small quantum cohomology ring of Grassmannians. The motivation of these combinatorial connections is the problem of finding a combinatorial/rep theoretic description of Gromov-Witten invariants.