ALGB Seminar: Alessandro Paolini, TU Kaiserslautern

Title: On the Decomposition Numbers of Finite Groups of Lie Type

Abstract:
Let G be a finite group, and let l be a prime number dividing |G|. The l-decomposition numbers of G relate the complex representations of G with its modular representations over a field of characteristic l. The goal of my talk is to survey recent techniques and results related to the determination of the decomposition numbers of a finite group of Lie type G(p^f) defined over a field with p^f elements when p >< l. I will first present recent progress towards the solution of a conjecture of Geck and Hiss about the unitriangularity of the decomposition matrices of
G(p^f) when p is a good prime. I will then explain some of the methods employed when p is a bad prime, with focus on the most recent results in this direction about the decomposition numbers of SO^+_8(2^f).