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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

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