How to obtain real irreducible representation matrices for finite point groups? I would like to generate the irreducible representation matrices in real (not complex) form for any finite point group, in order to use them in a projection operator. 
At least I require the diagonal elements (not its sum) of the representations.
does any one have a clue of how to obtain them?, a reference or whatever is welcome.
Thanks in advance.
 A: Here is an example of using Magma to get representations of $A_5$. Unfortunately this is not working in the Magma calculator. I think this is because it is trying to look up some data from a database, and the calcualtor does not have access to this. I will report the problem.
> G := AlternatingGroup(5);
> I := IrreducibleModules(G,Rationals());
> I;
[
    GModule of dimension 1 over Rational Field,
    GModule of dimension 4 over Rational Field,
    GModule of dimension 5 over Rational Field,
    GModule of dimension 6 over Rational Field
]
> r := Representation(I[4]);
> r( G!(2,3)(4,5) );
[-1  0  0  0  0  0]
[ 0  0  0  0  0 -1]
[ 0  0  0  0 -1  0]
[ 0  0  0 -1  0  0]
[ 0  0 -1  0  0  0]
[ 0 -1  0  0  0  0]
> r( G!(2,4,5) );
[ 0  0  0  0  1  0]
[ 0  0  0  1  0  0]
[ 0 -1  0  0  0  0]
[ 0  0 -1  0  0  0]
[ 0  0  0  0  0  1]
[ 1  0  0  0  0  0]

A: At least for icosahedral groups (I, Ih), I found the real matrix representations  documented in the papers:


*

*Pramana – J. Phys., Vol. 79, No. 6, 1365–1373 (DOI:10.1007/s12043-012-0334-1)

*J. Chem. Inf. Comput. Sci. 2003, 43, 1763-1770 (DOI:10.1021/ci025614x)

*Superlattices and Microstructures,Volume 3, Issue 4,1987, 391-398 (DOI:10.1016/0749-6036(87)90212-6)


3 years ago I did generate the real tables by myself by means of realizing a similarity transformation of the complex tables generators of Altmann and Herzig though. And I just very recently found the 1st and 2nd papers by chance :'D
