**CHEVIE** is a joint project of Meinolf Geck, Gerhard Hiss, Frank
Lübeck, Gunter Malle, Jean Michel, and Götz
Pfeiffer. We document here the development version 4 of the **GAP3**-part of
**CHEVIE**. This is a package in the **GAP3**3 language, which implements

• algorithms for: finite complex reflection groups and their cyclotomic Hecke algebras, arbitrary Coxeter groups, the corresponding braid groups, Kazhdan-Lusztig bases, left cells, root data, unipotent characters, unipotent and semi-simple elements of algebraic groups, Green functions, etc...

• contains library files holding information for finite complex reflection groups giving conjugacy classes, fake degrees, generic degrees, irreducible characters, representations of the associated Hecke algebras, associated unipotent characters and unipotent classes (for Weyl groups, or more generally, "Spetsial" groups).

The package is automatically loaded if you use the **GAP3** distribution
`gap3-jm`

; otherwise, you need to load it using

gap> RequirePackage("chevie"); --- Loading package chevie ------ version of 2018 Feb 19 ------ If you use CHEVIE in your work please cite the authors as follows: [Jean Michel] The development version of the CHEVIE package of GAP3 Journal of algebra 435 (2015) 308--336 [Meinolf Geck, Gerhard Hiss, Frank Luebeck, Gunter Malle, Goetz Pfeiffer] CHEVIE -- a system for computing and processing generic character tables Applicable Algebra in Engineering Comm. and Computing 7 (1996) 175--210

Compared to version 3, it is more general. For example, one can now work systematically with arbitrary Coxeter groups, not necessarily represented as permutation groups. Quite a few functions also work for arbitrary finite groups generated by complex reflections. Some functions have changed name to reflect the more general functionality. We have kept most former names working for compatibility, but we do not guarantee that they will survive in future releases.

Many objects associated with finite Coxeter groups admit some canonical
labeling which carries additional information. These labels are often
important for applications to Lie theory and related areas. The groups
constructed in the package are permutation or matrix groups, so all the
functions defined for such groups work; but often there are improvements,
exploiting the particular nature of these groups. For example, the generic
**GAP3** function `ConjugacyClasses`

applied to a Coxeter group does not
invoke the general algorithm for computing conjugacy classes of permutation
groups in **GAP3**, but first decomposes the given Coxeter group into
irreducible components, and then reads canonical representatives of minimal
length in the various classes of these irreducible components from library
files. These canonical representatives also come with some additional
information, for example the class names in exceptional groups reflect
Carter's admissible diagrams and in classical groups are given in terms of
partitions. In a similar way, the function `CharTable`

does not invoke the
Dixon--Schneider algorithm but proceeds in a similar way as described
above. Moreover, in the resulting character table the classes have the
labels described above and the characters also have canonical labels, e.g.
partitions of *n* in the case of the symmetric group * S_{n}*, which is
also the Coxeter group of type

Thus, most of the disk space required by **CHEVIE** is occupied by the files
containing the basic information about the finite irreducible reflection
groups. These files are called `weyla.g`

, `cmplxg24.g`

etc. up to the
biggest file `cmplxg34.g`

whose size is about *660* KBytes. These data
files are structured in a uniform manner so that any piece of information
can be extracted separately from them. (For example, it is not necessary to
first compute the character table in order to have labels for the
characters and classes.)

Several computations in the literature concerning the irreducible
characters of finite Coxeter groups and Iwahori--Hecke algebras can now be
checked or re-computed by anyone who is willing to use **GAP3** and
**CHEVIE**. Re-doing such computations and comparing with existing tables
has helped discover bugs in the programs and misprints in the literature;
we believe that having the possibility of repeating such computations and
experimenting with the results has increased the reliability of the data
and the programs. For example, it is now a trivial matter to re-compute the
tables of induce/restrict matrices (with the appropriate labeling of the
characters) for exceptional finite Weyl groups (see Section
ReflectionSubgroup). These matrices have various applications in the
representation theory of finite reductive groups, see chapter 4 of
Lusztig's book Lus85.

We ourselves have used these programs to prove results about the existence
of elements with special properties in the conjugacy classes of finite
Coxeter groups (see GP93, GM97), and to compute character
tables of Iwahori--Hecke algebras of exceptional type (see Gec94,
GM97). For a survey, see also Chv96. Quite a few computations
with finite complex reflection groups have also been made in **CHEVIE**.

*•* The user should observe limitations on storage for working with
these programs, e.g., the command `Elements`

applied to a Weyl group of
type *E _{8}* needs a computer with 360GB of main memory!

*•* There is a function `InfoChevie`

which is set equal to the **GAP3**
function `Ignore`

when you load **CHEVIE**. If you redefine it by
`InfoChevie:=Print;`

then the **CHEVIE** functions will print some
additional information in the course of their computations.

Of course, our hope is that more applications will be added in the future!
For contributions to **CHEVIE** have created a directory `contr`

in which
the corresponding files are distributed with **CHEVIE**. However, they do
remain under the authorship and the responsibility of their authors. Files
from that directory can be read into **GAP3** using the command
`ReadChv("contr/filename")`

. At present, the directory `contr`

contains
the following files:

`affa`

by F. Digne:- it contains functions to work with periodic
permutations of the integers, with the affine Coxeter group of type
seen as a group of periodic permutations, and with the corresponding dual Garside monoid.^{~}A

`arikidec`

by N. Jacon:- it contains functions for computing the canonical
basis of an arbitrary irreducible integrable highest weight representation
of the quantum group of the affine special linear group
*U*. It also computes the decomposition matrix of Ariki-Koike algebras, where the parameters are power of a e-th root of unity in a field of characteristic zero._{v}(\widehatsl_{e})

`braidsup`

by J. Michel:- it contains some supplementary programs for working with braids (or more generally Garside monoids).

`brbase`

by M. Geck and S. Kim:- it contains programs for computing bi-grassmannians and the base for the Bruhat--Chevalley order on finite Coxeter groups (see GK96).

`chargood`

by M. Geck and J. Michel:- it contains functions (used in
GM97) implementing algorithms to compute character tables of
Iwahori--Hecke algebras, especially that of type
*E*._{8}

`chevlie`

by M. Geck:- it contains function to build matrices for the Lie algebra and the Chevalley group attached to a root datum.

`cp`

by J. Michel and G. Neaime:- it contains a function to construct the Corran-Picantin monoid as an interval monoid, following Neaime's work.

`hecbloc`

by M. Geck:- it contains functions for computing blocks and defects of characters of Iwahori--Hecke algebras specialized at roots of unity over the rational numbers.

`minrep`

by M. Geck and G. Pfeiffer:- it contains programs (used in GP93) for computing representatives of minimal length in the conjugacy classes of finite Coxeter groups.

`murphy`

by A. Mathas:- it contains programs which allow calculations with
the Murphy basis of the Hecke algebra of type
*A*.

`rouquierblockdata`

by M. Chlouveraki and J. Michel:- it contains functions to compute the Rouquier blocks of 1-cylotomic Hecke algebras of arbitrary complex reflection groups.

`specpie`

by M. Geck and G. Malle:- it contains functions for computing the
Green-like polynomials (or rather rational functions) associated with
special pieces of the unipotent variety (or ``special'
`characters in the case of complex reflection groups'`

).

`spherical`

by D. Juteau:- it contains a function to determine the support
of the spherical module for a rational Cherednik algebra. Here is an
example:
gap> DisplaySphericalCriterion(ComplexReflectionGroup(13)); Maximal parabolic subgroups | q-index ______________________________________________________________________ A1 [ 2 ] |P2(x_1)P3(y_1)P2(x_1y_1)P2^2P6(x_1y_1^2) A1 [ 1 ] | P2P3(y_1)P2(x_1y_1)P2^2P6(x_1y_1^2)

`xy`

by J. Michel and R. Rouquier:- it contains a function to display graphically elements of Hecke modules for affine Weyl groups of rank 2.

Finally, it should be mentioned that the tables of Green functions for
finite groups of Lie type which are in the **MAPLE**-part of **CHEVIE** are
now obtainable by using the **CHEVIE** routines for unipotent classes and
the associated intersection cohomology complexes.

** Acknowledgments.** We wish to thank the Aachen **GAP3** team
for general support.

We also gratefully acknowledge financial support by the DFG in the framework of the Forschungsschwerpunkt "Algorithmische Zahlentheorie und Algebra" from 1992 to 1998.

We are indebted to Andrew Mathas for contributing the initial version of functions for the various Kazhdan-Lusztig bases in kl.g.

Index

gap3-jm

24 Apr 2021