# 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.

• By finite point group, do you just mean a finite group? – Tobias Kildetoft Jun 16 '15 at 8:58
• No. I mean a sub group of the orthogonal group O(3) that does not move at least one point and that does not contain infinitesimal rotations. here that wikipedia says: en.wikipedia.org/wiki/Point_groups_in_three_dimensions But maybe the method for getting the real irreducible representations matrices is the same in both cases (finite group or finite point group). :P – Horse time Jun 16 '15 at 9:06
• The cyclic and dihedral groups are not difficult - their real representations all have dimensions $1$ or $2$. Magma can compute matrices for the irreducible rational representations of any finite group. As far as I can see, for the isoloated examples, like $S_4$, $A_5$, $S_4 \times C_2$, $S_5 \times C_2$, the real representations are all rational anyway, so they could all be computed. – Derek Holt Jun 16 '15 at 15:10
• oh, I didn't know about the software Magma until now. I just notice that there is an online calculator magma.maths.usyd.edu.au/calc. I guess it is possible to use it to get the representations. As a example for $A_5$, how will be the input to get the real irreducible representation matrices? – Horse time Jun 16 '15 at 23:59

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);
> 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]

• wow :O ,that is pretty cool. It is a shame that it can't be reproduced in the online calculator. I'm seeing that the matrix dimension of the example is 6, but if it is irreducible, it should not be at most of dimension 5?, maybe it so because the rational restriction. in that case, I guess it is possible in that program to build the matrices including the irrational numbers and maybe get a lower dimension of the irreducible representations. – Horse time Jun 26 '15 at 21:27
• Very thanks by your help. I've just found a paper (little oldie) where the real irreducible representations (IR) matrices for $A_5$ (the icosahedral group) are some how tabulated sciencedirect.com/science/article/pii/0749603687902126# . However the work to retype those about 1200 matrices (for $I_h$ PG) would be very cumbersome, that's why it would be a better idea to generate them, for any finite PG, at least for the $A_5$, or instead to get the matrices in a file which can be readed with awk or something, (I could obtain that file through the use of Magma if it were free xD). – Horse time Jun 26 '15 at 21:59

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

1. Pramana – J. Phys., Vol. 79, No. 6, 1365–1373 (DOI:10.1007/s12043-012-0334-1)
2. J. Chem. Inf. Comput. Sci. 2003, 43, 1763-1770 (DOI:10.1021/ci025614x)
3. 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