Type: Seminar

Modified: Tue 4 Feb 2020 1743

Distribution: World

Expiry: 2 Feb 2021

CalTitle1: Gibson - How to compute and visualise characters of modular representations of rank 2 groups

Auth: romanova@10.17.126.222 (arom8272) in SMS-WASM

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/