Heaps, crystals, and preprojective algebra modules

TL;DR

This paper establishes a new explicit bijection between combinatorial and geometric models of certain crystal bases associated with simply-laced semisimple Lie algebras, using Nakajima's quiver varieties.

Contribution

It introduces a novel explicit bijection between height n reverse plane partitions and quiver Grassmannian components for crystal bases.

Findings

Bijection between combinatorial and geometric models of crystals

Proof that the bijection is an isomorphism of crystals

Application of Nakajima's tensor product quiver varieties

Abstract

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.