An Atlas of Modular Representation Theory, Version 1: Information on $\mathrm{Ext}^1$ for simple modules for groups of Lie type in defining characteristic over small fields

TL;DR

This paper compiles data on small-rank Lie type groups over small fields, focusing on Ext^1 between simple modules, to aid subgroup structure analysis of exceptional groups of Lie type.

Contribution

It provides the first comprehensive collection of Ext^1 data for simple modules of small-rank Lie groups over small fields, aiding representation theory research.

Findings

Ext^1 data for simple modules is almost complete

Includes dimensions of simple and Weyl modules

Lays groundwork for a detailed module structure database

Abstract

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, one may expect to include details about the socle structure of the projective modules, Jordan block structure of the action of unipotent elements, decompositions of symmetric and exterior powers of simple modules, and tensor products of modules, traces of semisimple elements and so on. The ideal place for such information is a dedicated website, connected to a database that could be queried to produce the information required. This document, while imperfect, will have to suffice for now.