Virtual Workshop on Recent Developments in Geometric Representation Theory

(work in progress with R. Rouquier) I will present a computational (yet conjectural) method to determine some decomposition matrices for finite groups of Lie type. These matrices encode how ordinary representations decompose when they are reduced to a field with positive characteristic $\ell$. There is an algorithm to compute them for $GL(n,q)$ when $\ell$ is large enough, but finding these matrices for other groups of Lie type is a very challenging problem.

In this talk I will focus on the finite general unitary group $GU(n,q)$. I will first explain how one can produce a "natural" self-equivalence in the case of $GL(n,q)$ coming from the topology of the Hilbert scheme of the complex plane. The combinatorial part of this equivalence is related to Macdonald's theory of symmetric functions and gives $(q,t)$-decomposition numbers. The evidence suggests that the case of finite unitary groups is obtained by taking a suitable square root of that equivalence, which encodes the relation between $GU(n,q)$ and $GL(n,-q)$.