81 The CHEVIE Package Version 4 -- a short introduction

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 GAP33 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 Sn, which is also the Coxeter group of type An-1 (see ChevieClassInfo and ChevieCharInfo). The normal forms we use, and the associated labeling of classes and characters for the individual types, are explained in detail in the various to chapters. The same considerations extend to some extent to all finite complex reflection groups.

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 E8 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 ~ A seen as a group of periodic permutations, and with the corresponding dual Garside monoid.

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 Uv (\widehatsle). 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.

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 E8.

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').

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.

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gap3-jm
18 Jun 2018