Find the conjugating matrix that can convert all the generators of a matrix group to simple integer matrices simultaneously.

[hongyi-zhao]

Hi Developers of RepnDecomp,

This is a problem similar to the one discussed here. This time, I want to find the conjugating matrix that can convert all the generators of a matrix group to simple integer matrices simultaneously, just like the TMPG229meToITA matrix as shown below:

gap> SGGenSet229me:=[
>   [[0, -1, 0, 1/4],
>   [0, 0, -1, 1/4],
>   [-1, 0, 0, 1/4],
>   [0, 0, 0, 1]],
>   [[-1, 2, -1, -3/8],
>   [-3/2, 3/2, -1/2, -1/16],
>   [-1/2, 3/2, -3/2, -17/16],
>   [0, 0, 0, 1]]
>   ];;
gap> TMSG229meToITA:=[[1, 1/2, 1/2, 1/8],
> [1/2, 1, 1/2, -3/8],
> [1/2, 1/2, 1, 1/8],
> [0, 0, 0, 1]];;
gap> PGGenSet229me:=SGGenSet229me{[1..Length(SGGenSet229me)]}{[1..3]}{[1..3]};;
gap> TMPG229meToITA:=TMSG229meToITA{[1..3]}{[1..3]};;
gap> List(PGGenSet229me, x-> TMPG229meToITA^-1 * x * TMPG229meToITA);
[ [ [ 0, -1, 0 ], [ 0, 0, -1 ], [ -1, 0, 0 ] ], [ [ 0, 1, 0 ], [ -1,
0, 0 ], [ 0, 0, -1 ] ] ]

I'm not sure whether RepnDecomp package is helpful in solving this problem. Any tips will be appreciated.

N.B.: This question has also been posted here.

Regards, Zhao

[kaashif]

I'm not actually a mathematician, and my linear algebra is pretty rusty so I'm not sure either. I'll think about it, but I don't immediately know whether any of the algorithms here can be adapted to produce simple matrices rather than probably-not-simple block diagonal matrices.

I'm pretty sure you'll get more useful responses from the people on the GAP mailing list.

[hongyi-zhao]

Thank you for your reply and comment.