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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
Representation Theory · Submitted 10 Mar 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | contribution 0.95 | 0.64 | 0.06 | asserted 1.00 |
| s1 | This information is important in the author's work on subgroup structure of exceptional groups of Lie type. | background 0.91 | 0.26 | 0.05 | asserted 1.00 |
| s2 | 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. | limitation 0.61 | 0.30 | 0.23 | asserted 1.00 |
| s3 | In addition, a lot of information about the dimensions of the simple and Weyl modules is included. | contribution 0.97 | 0.56 | 0.05 | asserted 1.00 |
| s4 | 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. | future work 0.98 | 0.09 | 0.04 | hedged possibility 0.98 |
| s5 | The ideal place for such information is a dedicated website, connected to a database that could be queried to produce the information required. | future work 0.98 | 0.06 | 0.04 | asserted 0.54 |
| s6 | This document, while imperfect, will have to suffice for now. | limitation 0.99 | 0.14 | 0.08 | asserted 0.93 |
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The paper
Original 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.
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