74 CHEVIE Version 4 -- a short introduction

CHEVIE is a joint project of Meinolf Geck, Gerhard Hiss, Frank Lübeck, Gunter Malle, Jean Michel, and Götz Pfeiffer. The GAP--part of CHEVIE is a package, written entirely in the GAP language, which implements

• algorithms for: (finite) complex reflection groups, Coxeter groups, the corresponding braid groups and Iwahori-Hecke algebras, cyclotomic Hecke algebras, root systems, Kazhdan-Lusztig polynomials, lefts cells, etc...

• contains library files holding information for finite complex reflection groups giving conjugacy classes, fake degrees, generic degrees, irreducible characters, representations of the groups and the associated Hecke algebras. The package is loaded using the command RequirePackage("chevie") (see RequirePackage). 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 group 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 constructed in or 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 GAP function ConjugacyClasses applied to a Coxeter group does not invoke the general algorithm for computing conjugacy classes of permutation groups in GAP, 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, the resulting character table labels classes as above and also labels the characters, with e.g. partitions of n in the case of the symmetric group \mathfrak Sn, i.e., the Coxeter group of type An-1 (see ChevieClassInfo and ChevieCharInfo). The conventions that we use about normal forms of elements, labeling of classes and characters for the individual types are explained in detail in the various to chapters.

The same is true to some extent with complex reflection groups. Thus, most of the disk space required by the CHEVIE files 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 weyle8.g whose size is about 430 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 GAP and CHEVIE. Re-doing such computations and comparing with existing tables has sometimes lead to the discovery of bugs in the programs or to 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 E8 will cause every computer yet designed to run out of memory!

There is a function InfoChevie which is set equal to the GAP 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 from outside (or one or several among us) we 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 GAP using the command ReadChv("contr/filename"). At present, the directory contr contains the following files:

murphy by A. Mathas; it contains programs which allow calculations with the Murphy basis of the Hecke algebra of type A.

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.

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).

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

conjbrai by N. Franco; it contains programs which solve the conjugacy problem and compute centralizers in braid (or more generally Garside) groups.

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 E8.

Finally, it should be mentioned that there is also a MAPLE-part of CHEVIE which contains generic character tables of finite groups of Lie type and tables of Green functions. The conventions about data related to the associated finite Weyl groups are compatible with those in the present package. It is planned that, in the not too far future, the MAPLE-part will be re-written in GAP.

Acknowledgments. We wish to thank the Aachen GAP team for general support over the last years.

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

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

Paris, Kassel and Aachen, 2002

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GAP 3.4.4
April 1997