Heaps, crystals, and preprojective algebra modules

Fix a simply-laced semisimple Lie algebra. We study the crystal $B(n\lambda )$, were $\lambda$ is a dominant minuscule weight and $n$ is a natural number. On one hand, $B(n\lambda )$ can be realized combinatorially by height $n$ reverse plane partitions on a heap associated to $\lambda$. On the other hand, we use this heap to define a module over the preprojective algebra of the underlying Dynkin quiver. Using the work of Saito and Savage-Tingley, we realize $B(n\lambda )$ via irreducible components of the quiver Grassmannian of $n$ copies of this module. In this paper, we describe an explicit bijection between these two models for $B(n\lambda )$ and prove that our bijection yields an isomorphism of crystals. Our main geometric tool is Nakajima’s tensor product quiver varieties.