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
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
cmplxg24.g etc. up to the
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
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
the following files:
affaby F. Digne:
arikidecby N. Jacon:
braidsupby J. Michel:
brbaseby M. Geck and S. Kim:
chargoodby M. Geck and J. Michel:
chevlieby M. Geck:
cpby J. Michel and G. Neaime:
hecblocby M. Geck:
minrepby M. Geck and G. Pfeiffer:
murphyby A. Mathas:
rouquierblockdataby M. Chlouveraki and J. Michel:
specpieby M. Geck and G. Malle:
characters in the case of complex reflection groups').
sphericalby D. Juteau:
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)
xyby J. Michel and R. Rouquier:
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.