Crystals, Heaps and preprojective algebra modules

Abstract

A crystal is a colored directed graph associated to a representation of a reductive algebraic group that enables us to understand the structure of the representation in purely combinatorial terms. Crystals are very easy and fun to work with. In the first hour we will give a hands-on introduction to crystals with lots of examples. Then we will study a concrete type independent model for certain special crystals based on heaps, which are configurations of beads on runners.

In the second hour we will shift to geometry and introduce a second model for crystals on the irreducible components of a certain quiver variety. Then we'll see how the Jordan canonical form provides a crystal isomorphism between the two models. This is joint work with Anne Dranowski, Joel Kamnitzer and Calder Morton-Ferguson.