Introduction to Categorification, Diagrammatics and the Hecke Category

Contact for this talk: Alexander Thomas (athomas@mpim-bonn.mpg.de)

The Hecke category is a ubiquitous object in representation theory. It can be constructed using sheaves on the flag variety, or certain infinite-dimensional representations (in the Bernstein-Gelfand-Gelfand category O). It acts on rational representations of algebraic groups, and on other categories in modular representation theory. However, a modern perspective is to describe the category by generators and relations, which has many advantages: it allows computers and humans (even graduate students) to perform calculations, it gives access to finite characteristic settings, it permits easier constructions of actions, etcetera. The main drawback to generators and relations is not for the end-user but for the originator: it is not easy to prove that a presentation is correct. For example, there are many non-trivial presentations of the trivial group.

This talk will begin by discussing some of the philosophy of categorification and its important shadow, the Hom form. Then we introduce the Kazhdan-Lusztig problem, to motivate the question of finding a presentation for the Hecke category.  First we'll address an easier problem: finding a presentation for the 2-groupoid of the symmetric group, using this as an opportunity to introduce diagrammatic methods. After the break, we'll discuss how the presentation of the Hecke category was found, and also how it was proven, in order to help those who might wish to follow a similar path.

This talk is on work (joint with Khovanov and Williamson) much of which was originally done at the Max Planck Institute in Bonn over a decade ago.