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Dear Gap-Forum,

Dima Pasechnik wrote

we need to decompose G-modules (for G being

a group of order say < 10^5) of dimension up to few hundred

over Z (or over an extension of Z, like Z[i]).

While we can perfectly accomplish the tack modulo a prime,

(or more generally, over a GF(q))

using GAP's MTX package, doing this over Z is harder, and I

am not aware of any algorithms that can do this automatically,

leave alone something implemented in GAP for this purpose.I would most appreciate any references and pointers to tools

that can help us.

There is a paper by Plesken and Souvignier

Plesken, Wilhelm(D-AACH-BM); Souvignier, Bernd(D-AACH-BM)

Constructing rational representations of finite groups. (English.

English summary)

Experiment. Math. 5 (1996), no. 1, 39--47.

which is related to this problem.

Plesken also had a student Tilman Schultz who worked on the related

problem of finding all of the integral representations of a given group.

He wrote some programs, but I don't beleive that they ever got to the

push-button stage - I think they required some intelligent intervention by

the user.

I have a manuscript by Richard Parker called 'An integral meataxe' -

unfortunately I cannot find the electronic source of this. I am fairly

sure that he wrote some programmes along these lines. He might be the

best hope for a working program.

Also, I know that they are working on this problem in Sydney, and have lots

of bits of code, but I don't believe that any of this has been released in

Magma yet.

The problem is not an easy one. The sum of isomorphic irreducibles can be

very difficult to decomposes. When the consituents are not isomorphic,

it is not so hard to decompose the module, but there is another problem,

that the integral or rational entries in the matrices you are working with

start getting out of control very quickly with each decomposition, so

you need to use something like LLL to keep the entries small.

Derek Holt.

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