Atlas of Lie Groups and Representations

Software

The atlas software

The Atlas of Lie Groups and Representations software does computations with representations of real reductive groups, from basic structure theory through computing unitary representations. It is free and open source, and runs on Linux, Mac and Windows.

Download and install Source code on GitHub Documentation

What the software does

The software allows the user to define an arbitrary reductive group, and gives access to its root data. The user can also define an arbitrary real form G of such a group. The user can define a parameter for an arbitrary irreducible admissible representation of G, and the software will compute its block, Cayley transforms, cross actions, and Kazhdan–Lusztig–Vogan polynomials. In particular this includes character formulas (for irreducible representations) and composition series (for standard modules).

The software computes the signature of the invariant Hermitian form on an irreducible representation π, and therefore can determine if π is unitary. See the mathematical background, and for complete details the paper Unitary representations of real reductive groups by Jeffrey Adams, Peter Trapa, Marc van Leeuwen and David A. Vogan, Jr.

More detail: capabilities of the software (as of version 1.0, February 2017) and the documentation of the atlas library.

Main features

Also on its own page: summary of features.

The basics

  • Complex reductive groups, root systems
  • Real reductive groups, inner classes and forms
  • Maximal compact subgroup K(ℝ) and its complexification K
  • K-orbits on G/B
  • Langlands parameters
  • Irreducible representations of K
  • Branching to K
  • Kazhdan–Lusztig–Vogan polynomials
  • Composition series and character formulas

Hermitian forms and unitarity

  • Hermitian representations, invariant Hermitian forms
  • The c-invariant form
  • Signatures of forms
  • Unitarity
  • Jantzen filtration

Induction

  • Parabolic subgroups, Levi subgroups
  • Real parabolic induction and cohomological induction

Structure theory

  • Levi subgroups
  • Pseudo-Levi subgroups
  • Equal rank reductive subgroups and their poset

The Weyl group

  • The Weyl group W
  • Conjugacy classes, characters of W
  • Character table of W
  • Representations of W
  • Coherent continuation

Nilpotent orbits

  • Nilpotent orbits of a complex group
  • Nilpotent orbits of a real group
  • Component group of the centralizer of a nilpotent element
  • Special nilpotent orbits
  • Duality of nilpotent orbits
  • Springer correspondence
  • Associated variety of ideals in the universal enveloping algebra
  • Gelfand–Kirillov dimension

Lusztig’s theory of the Weyl group

  • Generic degree and fake degree
  • Cells of representations of the Weyl group
  • Lusztig’s parametrization of representations of W

Unipotent representations

  • Unipotent representations of a real reductive group
  • Weak Arthur packets of unipotent representations

Miscellaneous

  • Structure constants
  • The Tits group
  • Extended groups

Download and install

The source code is on GitHub at github.com/jeffreyadams/atlasofliegroups. Compiling it from source is the preferred method. The software is written in C++ and needs a C++ compiler.

Compile from source

git clone https://github.com/jeffreyadams/atlasofliegroups.git
cd atlasofliegroups
make

This produces the executable atlas. make optimize=true is recommended: the compilation is slower, but the code runs substantially faster. make install makes atlas accessible from anywhere (it puts a shell script in ~/bin), and atlas then loads the scripts in atlas-scripts (the file all.at) automatically. To update later, run git pull origin master in the atlasofliegroups directory (and make again if any source files changed).

Run it with Docker

The next choice is the Docker container system: the image jeffreyadams/atlasofliegroups runs the software in a self-contained Linux environment, so it depends less on the details of your system. This is a good option if you have trouble compiling the software yourself.

Step-by-step guides

Old release archives (versions 1.0 to 1.0.8) and install pages from 2012–2016 are kept in the download archive. They are out of date.

Learning to use it

The online web interface to the software has been shut down; see web interface.

History

The original software was written by Fokko du Cloux (1954–2006). In November 2006 Marc van Leeuwen took over primary responsibility for the code.

Jan 2016
Version 0.6. What used to be called atlas is now Fokko, and what used to be called realex is now atlas.
Feb 2017
Version 1.0.
Dec 2021
Version 1.1.
Dec 2024
Version 1.2 (source code on GitHub).

Compiling the software also produces the program Fokko, which has the core software but not the scripting language. Fokko du Cloux’s Coxeter software, for computing Kazhdan–Lusztig polynomials for Coxeter groups, is also available on this site.

License

The atlas software is free software, distributed under the GNU General Public License version 3. See copyright and license. For related software and data elsewhere, see related sites.