> < ^ Date: Mon, 23 Sep 2002 17:46:54 +0200
> < ^ From: Thomas Breuer <Thomas.Breuer@Math.RWTH-Aachen.DE >
< ^ Subject: Re: computing certain parts of a character table

Dear GAP Forum,

Computing the character table of a large group may, of course, take a very
long time. But if I'm just interested in a few characters (like, for
instance, the irreducible characters of degree n for some fixed n or the
faithful irreducible characters), if there some faster way to compute
them than computing the whole character table?

There are methods that compute new characters from old, but often it is not
guaranteed that thay will obtain all characters. What you can try is to run
the Dixon-Schneider Algorithm for a few sets (see section `Advanced Methods
for the Dixon-Schneider algorithm' of the reference manual:
http://www.gap-system.org/Manual4/htm/ref/CHAP069.htm#SECT014
)
and then try toi obtain the remaining characters by working with them as
class functions (see chapter ``Class functions'' in the manual), however it
is perceivable that this process will not find all characters you are
interested in, but only a subset. In particular it is hard to guarantee that
one has found all characters of a given degree, or all faithful characters,
unless the whole table has been determined.

It depends on the structure of the group what additional tricks are
available:

If the group is solvable then one can compute the irreducible degrees
with reasonable effort; from that one knows whether one has found
all irreducibles of a given degree.

If the group is abelian-by-supersolvable then one should try to compute
the irreducible characters with the Baum-Clausen method instead of Dixon's
algorithm.

The computations can be accelerated by choosing an appropriate
representation of the group, e.g., a PC group instead of a permutation
group if possible.

If the group has a small nontrivial normal subgroup then one can compute
the characters of the factor group first, lift them to characters
of the whole group, and feed them into the computation of the remaining
characters.

And if related information is stored in the character table library then
one can of course use it.

All the best,
Thomas

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