[GAP Forum] Conjugacy in GLn(Z)

[Alexander Hulpke]

Dear Forum, Dear Mathieu,

> I have a set of matrices in GLn(Z) and I want to reduce them
> by conjugacy. The problem is that the existing tools in GAP
> seem too slow.

[...]
> 
> Two general strategy are available in GAP
> 1> Use
>   RepresentativeAction(GL(8, Integers), g1, g2)

This will start an orbit calculation on g1, trying to find g2. Thus not only will it be slow to succeed, it will not terminate if they are not conjugate.

> 2> Use carat by computing the Bravais groups
>   of Gi = BravaisGroup( Group( [gi] ) ), testing
>   for conjugacy, computing the normalizer and then
>   concluding.

This is probably the best possible with existing code.

GAP doesn't really have much built-in for infinite matrix groups, in particular none of the methods you allude to have been implemented for GL over the integers.

If you wanted to implement it yourself,
http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/matrixconj.pdf
might be a good starting point, but it in the end becomes a problem in number theory and the lack of ideal class functions would be inconvenient.
Your best bet might be to see what kind of conjugacy (either over larger ring, or not perfect) you could live with.

Sorry!

   Alexander