> < ^ Date: Wed, 05 Feb 2003 00:16:01 +0100 (CET)
> < ^ From: Frank Luebeck <frank.luebeck@math.rwth-aachen.de >
^ Subject: Series of character tables

Dear GAP-forum,

In a recent mail from M. Darafsheh we saw that from time to time there is
interest in character tables of groups in series whose representation theory
is well understood, but whose tables are not contained in the GAP library.
In that case the group L_8(2) = GL(8,2) was mentioned.

The GAP character table library could be blown up considerably by adding
series of tables which can be computed systematically. To avoid this only
tables which find some explicit interest are added upon request.

Concerning the series of general linear groups GL(n,q), q a prime power, the
status of computing explicit complex character values is as follows (to my
knowledge):

The complete character tables of GL(n,q) are known by the classical paper:

Green, J. A. The characters of the finite general linear groups. Trans.
Amer. Math. Soc. 80 (1955), 402--447.

Nevertheless, writing down a table for explicit values of n and q from that
combinatorial description is still somewhat challenging.

For example, this was worked out in:

Hemkemeier, B. and Jürgens, U. Hall-Polynome und Charaktere von
\$GL_n(q)\$. Diploma thesis, Universität Bielefeld (1992)

The authors also provided a computer program that computes each table of
GL(n,q) which has a reasonable size to be used on a computer. (It seems that
this program is no longer distributed but I have a version of it that can be
used to compute any such table upon request.)

This thesis is probably not widely known and I wouldn't be too surprised to
hear that there are other articles or theses on the question of using
Green's result for explicit computations of character values.

Furthermore there exist 'generic character tables' which give for fixed \$n\$
a parameterized description of the character values of the groups GL(n,q)
for all q at once. Explicit such tables were given by Jordan, Schur and
Steinberg for n <= 4. I have computed them for n=5,6 (using Deligne-Lusztig
theory which also yields the generic tables for the unitary groups GU(n,q)).
These tables exist in Maple readable format and can be used for computations
in the CHEVIE package. Partial tables can also be computed for bigger n.

If you are interested in any character table mentioned here that is not
already distributed just send a request to:

gap-trouble@dcs.st-and.ac.uk

(Similarly, if you need other tables of which you suspect that they are not
contained in the GAP library just because of space considerations.)

With best regards,

Frank Lübeck

```///  Dr. Frank Lübeck, Lehrstuhl D für Mathematik, Templergraben 64,  ///
\\\                    52062 Aachen, Germany                          \\\
///  E-mail: Frank.Luebeck@Math.RWTH-Aachen.De                        ///
\\\  WWW:    http://www.math.rwth-aachen.de/~Frank.Luebeck/           \\\
```

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