Dear GAP Forum members,

Barry Monson asked in the Forum:

Does GAP have the means to compute real irreducible
components, say for a group of reasonably small order and
representation degree (eg. a subgroup of order 128 in S_16)? (Maybe
that doesn't qualify as reasonable).

I realize that the "matrix" package can handle my group over a
finite field, and this has been of some use. Still, I want real
representations, or more accurately, representations over some
finite extension of the rationals.

I suppose this ultimately means finding common eigenvectors
for the generators of some algebra. However, I am really only
seeking to understand a few examples, so that I definitely want to
avoid writing my own program to cope with such things.

Given a real representation (e.g. a permutation representation of a
group of order 128, as suggested in the question) there are then three
interpretations of asking for its irreducible components.

If one wants just to know which irreducible representations are
constituents of the given one, of course character theory suffices.
You could use the Dixon/Schneider method to find the character table
of your group and then determine the decomposition of the character of
the given representation (see section 48 of the GAP manual).

The next step could be to ask just to compute matrix representations
equivalent to those constituents. For a p-group they can be found
using MatRepresentationsPGroup, see section 48.25. This actually works
for certain monomial groups, namely such that contain an abelian
normal subgroup with a supersoluble factorgroup.

There exists a more general method for constructing irreducible
characteristic zero representations of finite soluble groups which has
been proposed in a paper by Wilhelm Plesken 'Towards a soluble
quotient algorithm' (JSC 4) and which recently has been implemented by
Herbert Brueckner, a PhD student of Prof. Plesken, in the context of
a full implementation of the Plesken SQ. GAP contains only an older
implementation of parts of that proposal, in which just the part that
might be of interest for you is missing. If you want more information
about Brueckner's implementation and its availability you might
contact Prof. Plesken (plesken@willi.math.rwth-aachen.de). The
implememtation is in C, and said to be very efficient, but since it is
a standalone, not (yet?) accessible as a share package of GAP it might
be less comfortable to use than just calling a GAP function.

The ultimate request that you may have (and probably in fact have) is
to ask for the actual irreducible submodules of your given
representation, i.e. to ask for a matrix that will tranform the given
representation into block diagonal form with irreducible diagonal
blocks. Again to the best of my knowledge there is nothing in GAP that
could help you.

Having explicitely the irreducible matrix representations as you can
get them for the above mentioned monomial groups from GAP or for
soluble groups possibly from Brueckner's program one can actually
construct a 'symmetry adapted basis' (as the physicists say) using the
Schur relations. Although I do not know of a ready to use
implementation of this, it should be possible to implement this in
GAP.

Since there seems to be no counterpart of Plesken's method known for
insoluble groups, for these one would need an analogon of the meat-axe
for characteristic zero. Richard Parker has propagated the idea of
such an 'integral meataxe' (for arbitrary finite groups), and during a
visit to Aachen he has been working on an implementation of his
ideas. You should best contact Richard Parker (richardp@ukonline.uk, I
hope) or Prof. Plesken to get up to date information about the present
state of the realisation of those ideas.

Sorry that this time we aren't of better help, but perhaps the
suggested contacts may get you further.

Kind regards Joachim Neubueser