Remarks on modular representations of finite groups of Lie type in non-defining characteristic
Meinolf Geck
备注：
11 pages; this note corresponds to my talk at the 10th RAQ conference in Nagoya (August, 2010). It will appear in the proceedings, AMS Contemporary Mathematics
Let $G$ be a finite group of Lie type and $\ell$ be a prime which is not equal to the defining characteristic of $G$. In this note we discuss some open problems concerning the $\ell$-modular irreducible representations of $G$. We also establish a strengthening of the results in [13] on the classification of the $\ell$-modular principal series representations of $G$.
This article discusses the modular representation theory of finite groups of Lie type from the viewpoint of Broue's abelian defect group conjecture. We discuss both the defining characteristic case, the inspiration for Alperin's weight conjecture, and the non-defining case, the inspiration for Broue's conjecture. The modular representation theory of general finite groups is conjectured to behave both like that of finite groups of Lie type in defining characteristic, and in non-defining characteristic, to a large extent. The expected behaviour of modular representation theory of finite groups o…
This document is the first iteration of an attempt to collate information about small-rank groups of Lie type over small fields, and their representation theory over the defining field. This information is important in the author's work on subgroup structure of exceptional groups of Lie type. The most important information in that work is information about $\mathrm{Ext}^1$ between simple modules, and so in Version 1 of this document, that data is almost all of the data available. In addition, a lot of information about the dimensions of the simple and Weyl modules is included. More generally, …
We study properties of relative modular categories and derive sufficient conditions for their existence. In particular, we derive sufficient conditions for relative pre-modular categories to be non-degenerate and relative modular, and for the quotient of categories taking a particular form by their ideal of negligible modules to be generically semi-simple.
In this paper we introduce a novel family of Markov chains on the simple representations of $\mathrm{SL}_2(\mathbb{F}_p)$ in defining characteristic, defined by tensoring with a fixed simple module and choosing an indecomposable non-projective summand. We show these chains are reversible and find their connected components and their stationary distributions. We draw connections between the properties of the chain and the representation theory of $\mathrm{SL}_2(\mathbb{F}_p)$, emphasising symmetries of the tensor product. We also provide an elementary proof of the decomposition of tensor produc…
