Heaps, crystals, and preprojective algebra modules

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.

arXiv comment42 pages

References in corpus (6)

• Quiver varieties and tensor products
• Geometric and combinatorial realizations of crystal graphs
• Tensor product varieties and crystals. GL case
• Quiver grassmannians, quiver varieties and the preprojective algebra
• Verma modules and preprojective algebras
• Uniform description of the rigged configuration bijection

Cited by in corpus (1)

• q-Whittaker functions, finite fields, and Jordan forms