Heaps, Crystals and Preprojective Algebra Modules

A Kashiwara crystal is a combinatorial gadget associated to a representation of a reductive algebraic group that enables us to understand the structure of the representation in purely combinatorial terms. We will describe a type-independent combinatorial construction of crystals of the form $B_w(n\lambda)$, using the heap associated to a fully commutative element $w$ in the Weyl group. Then we will discuss how we can use the heap to also define a module for the preprojective algebra of the underlying Dynkin quiver. Using the work of Savage and Tingley, we also realize the crystal $B_w(n\lambda)$ via irreducible components of the quiver Grassmannians of n copies of this module, and we describe an explicit crystal isomorphism between the two models. This is joint work with Anne Dranowski, Joel Kamnitzer and Calder Morton-Ferguson.