# How to get real irreducible matrix representations from the complex irreducible matrix representations?

I'm trying to get real symmetry adapted orbitals for molecules with icosahedric symmetry (point groups $I$ and $I_h$) using the complete projector operator (truly projector if i=j): $$\hat{P}_{ij}^{ \Gamma_\ell} =\frac{d_\ell}{h} \sum_{R}^h R_{ij}^{\ell } \hat{R}$$ Here $R_{ij}^{\ell }$ points the element located in row $i$ and column $j$ of the irreducible matrix representation $\mathbf{R}^\ell$ corresponding to the irreducible representation labeled $\Gamma_\ell$ for the symmetry operator $\hat{R}$. $d_\ell$ is the dimension of $\mathbf{R}^\ell$ and $h$ is the group order. So I require the complete matrix representations and not only the characters. In page 655 of the officially free book Point-Group Theory Tables. Altmann and Herzig. 2nd Ed. 2011, there are tabulated generators of the irreducible (complex) matrix representations of these groups. Here arises the question: How to obtain the corresponding real (and orthogonal) irreducible matrices from the complex ones?.

Thanks in advance. Whatever kind of help is welcome. Greetings.

• For icosahedral symmetry the answer is here – Horse time Jan 24 '19 at 12:53