Informal Friday Seminar: Gibson -- How to compute and visualise characters of modular representations of rank 2 groups

In characteristic zero, the characters of representations of reductive groups have been 
understood for a long time and can be computed in many ways, two of the most famous 
being the Weyl and Demazure character formulas. In positive characteristic, these 
characters are far more difficult to understand, and a general procedure for computing 
them is still unknown. However, in small cases we can "get lucky" with a filtration of 
the Weyl modules due to Jantzen and compute the characters directly. I have written some 
software which carries out these computations for the rank 2 reductive groups and 
displays them visually, which works fast enough that one can actually "interact" with 
these characters and watch them change.

In this talk, I will (assisted by some visualisations) go through how these computations 
work, which involves understanding the weight lattice, the dot-action of both the Weyl 
and p-dialated affine Weyl groups, the Weyl/Demazure character formula, the Jantzen 
filtration of the Weyl module and its associated formula, and the Steinberg tensor 
product theorem. Towards the end of the talk I will show you (rather than tell you) 
about some interesting phenomena and repeating patterns in the characters of simple 
modules in the modular setting.

The link to the software I’m talking about: 
https://www.jgibson.id.au/articles/rank2reps/