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The ATLAS of Group Representations, V4

Welcome to the Atlas of Group Representations. This is Version 4 of the Atlas, started by David Craven of the University of Birmingham. It involves a complete restructuring and regeneration of the information in the previous version of the Atlas, which were created by Rob Wilson.

For more information about how to use this website, lists of known bugs and feature requests, and to suggest new features or bug report, see below.

The Groups

Alternating groups: contains $\mathrm{A}_\mathrm{n}$ up to $\mathrm{n}=13$, $\mathrm{S}_\mathrm{n}$ up to $\mathrm{n}=11$, and the other almost simple groups with socle $\mathrm{A}_6$.

Linear groups: contains $\mathrm{L}_2(\mathrm{q})$ up to $\mathrm{q}=32$ and their almost simple groups, $\mathrm{L}_3(\mathrm{q})$ up to $\mathrm{q}=9$, and a few larger-rank simple groups.

Unitary groups: contains very little at the moment, but this will change soon!

Symplectic groups: contains very little at the moment, but this will change soon!

Exceptional groups: contains very little at the moment, but this will change soon!

Sporadic groups: contains all of the smaller sporadic simple groups and almost simple groups, and a few of the larger ones.

Status of the Atlas V4

Version 4 of the Atlas is currently under development, both in terms of expanding the catalogue of groups and also the functionality of the website (which have been separated completely in this version). Because significantly more information is given in Version 4 (character tables, decomposition matrices, full lists of class representatives) one cannot simply port over the information from Version 3 but must rederive the whole thing from scratch.

List of known bugs

  1. I need to delete the new line before the >; in the Magma permutation representation output. Will do this very soon.
  2. The website is, for some reason, not working on Google mobile, at least on mine. I think this is an issue with the Table options accordion. Looking into it!
  3. Some of the options for downloading aren't coded yet. For example, Magma permutation reps, linear reps and maximal subgroup SLPs are fine, but GAP ones aren't coded yet. This is largely because I am not sure exactly how to present them in the GAP language. Will fix this!
  4. Characteristic 0 representations are largely not in the Atlas at the moment, because of issues getting integral representations and how to deal with irrationalities. Those that are, in the Magma output, are prefaced with "F:=GF(Z)" because Z is there rather than q. This will be fixed, but while there are very few representations in characteristic 0 it's on the back burner.
  5. Big tables do not have side scrolling so you cannot see them. I know, I know. Will deal with this. Also do it with modular character tables.
  6. Fix the breadcrumbs for the index page.

List of desirable features

  1. Allow the factorizations of the orders and indices of maximal subgroups and class centralizers to be displayed rather than the integers themselves. Obviously very useful for determining which subgroups contain Sylow subgroups, etc. The information has started to be included in the database, but I will need to write a script that adds it to the already created database entries and then modify the tables to allow this to be toggled.
  2. Make the class representatives field optional on the tables. They aren't really needed by humans, and take up a lot of room.
  3. Allow the class representatives to be downloaded as Magma/GAP programs for ease of making class reps. This is easy to do, but there is a clever way to make an SLPish type of situation, where you compute ab, store that, then multiply that by b say, and that appears in another class, and build up the classes ike this rather than evaluate each word. Important for some big groups like $\mathrm{Th}$ or $\mathrm{BM}$, or in big representations of groups where you want to create a Brauer character but you cannot enumerate the conjugacy classes directly, like for $\mathrm{J}_4$.
  4. Add the 1- and 2-cohomologies (if known) to the linear representations pages. Will do this soon, it's easy.
  5. Make the Navigate panel on the right embedded in the header and collapsed by default. Also make the header shrink/disappear when scrolling down the page.
  6. Create an option to allow Atlas labelling in the character table and reorder the rows of groups that are not almost simple. Expand this to allow the partition labelling for symmetric groups.
  7. Create an interpreter for the simpler Atlas irrationalities. I have such a thing, replacing r2 by $√2$ for example, but I need to make it better.
  8. At the bottom of each page, there should be a list of sources for the data, author data, and versioning. This is all in the database, butneeds to be written as a footer file.
  9. Add the permutation character to the table of permutation representations (or just the Details page?).
  10. Add information about elementary abelian subgroups and local information about the Sylow subgroups (order, rank, nilpotency class, sectional rank, exponent, etc.). This is not yet in the database so would have to be calculated for each group.

How to use this Atlas

This Atlas contains a lot more information than the old one, and so a bit of information is needed to get the most out of the new database. First, on ordinary and modular character tables rows and columns can be highlighted by clicking on their headers. Second, click on Table options to find out what other information exists but that isn't displayed by default. This also allows restricting by characteristic when viewing linear representations, for example. The variants drop-down menu on most pages allows you to easily move between the groups with the same simple composition factor, which can often be useful. Finally, the Details button of permutation representations gives you the cycle structure of the group classes on the underlying set, the rank and suborbit degrees, which might be of value to people.