Scott Morrison (Microsoft Station Q)

Friday 26th September, 12.05-12.55pm, Carslaw 159

Generators and relations for the representation theory of U_q(sl_n) as a planar algebra

I'll begin by explaining what a planar algebra is, and then show you how the representation theory of SU(2) and SU(3) can be given a "finite presentation by generators and relations" as a planar algebra. This may be familiar to some, as the Temperley-Lieb algebra for SU(2) or Kuperberg's spider for SU(3). Next, I'll explain my work on generalising this to all SU(n). We'll start with a category of diagrams, generated by some trivalent vertices, and a surjective map to the representation theory; the difficulty will be understanding the relations amongst these diagrams. The main trick is to remember than SU(n) sits inside SU(n+1), and conversely representations of SU(n+1) break up (or "branch") as representations of SU(n). I'll explain how to understand the combinatorics of branching in terms of my diagrams, and how to use this to "lift" relations for diagrams from one level to the next.