> < ^ Date: Tue, 30 Mar 1993 21:04:46 +0200
> ^ From: Peter Webb <webb@math.umn.edu >
^ Subject: Loewy series and representation theory

In connection with Thierry Dana-Picard's question about finding Loewy series,
I have written several GAP routines which handle group representations which
in particular will compute the Loewy series of any representation of a
p-group in characteristic p (and more generally the Loewy series of the
largest quotient of a module all of whose composition factors are trivial).
The algorithm I use is simply to multiply repeatedly by the augmentation
ideal. The trouble with this approach if one is interested in the Loewy
series of the group ring is that the regular representation has a large
dimension, and for reasons of time and storage the problem may become
computationally unfeasible that way. Jenning's theorem could be the better

On the topic of representation theory within GAP, I have the impression
that this side of things has been somewhat neglected so far. The meataxe
is implemented, but I have other goals in mind to do with creating software
to complement this. For example, the meataxe would not be so good for
analyzing the structure of modules for p-groups in characteristic p, but
algorithms based upon the computation of fixed points are very effective in
this situation. It is a long-term project for me to expand what software
I have, and to put it into a publicly acceptable state. Right now, for
example, it does not properly conform to the object-oriented style of GAP,
and it is not adequately tested. At this point I would be happy to hear of
others writing similar software (some I already know of). My general aim
is to have a package which computes Loewy series reasonably, will extract
a quotient in the Loewy series of a p-group as a representation of its
normalizer in a larger group (for example), will compute relative traces
between modules of fixed points, and such similar things.

Peter Webb

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