ALGB Seminar: Thomas Gerber, RWT Aachen

Title: Crystals and higher level Fock spaces

Abstract: In the 90s, Kashiwara introduced the notion of crystals for integrable representations of quantum groups. Encoded in a graph, the Kashiwara crystal reflects, at the combinatorial level, certain essential algebraic properties of the representation.
For a particular representation, the Fock space, I will explain the existence of a new type of crystal, arising from the action of the quantum Heisenberg algebra, and how to compute it explicitly.
Finally, I will mention how both types of crystals (Kashiwara and Heisenberg) are used to solve fundamental problems in modular representation theory (of reflection groups, Hecke algebras, and finite classical groups).