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.