How to generate the icosahedral groups $I$ and $I_h$?

Question

The icosahedral groups $I$ with 60 elements and $I_h = I \times Z_2$ are also three dimensional point groups. However, ever unlike other point groups, it seems there is rarely reference to give their representation (e.g., matrix representation for each elements). Can any one help me how to generate them?

Answer 1

A quick google search pulls up this paper by Litvin which includes permutation and matrix representations.

For example, it says $I$ is generated as a permutation group by the following elements of $S_{12}$ (and identifies them to more classical geometric generators, which I omit here for brevity): $$g1=(1,2) (3,6) (4,11) (5,7)(8, 10) (9,12)$$ $$g2=(1,4,3) (2,5,8) (6,9,7) (10,12,11)$$ $$g3=(1) (12) (2,6,5,4,3) (7,11,10,9,8),$$ and $I_h$ is generated by adding in the spatial inversion $$g4=(1,12) (2,9) (3,10) (4,11)(5,7) (6,8).$$ $3\times 3$ matrix representations are also provided for each of these generators.

They are (assuming I'm reading the $g3$ entry correctly): $$g1 = \begin{pmatrix} -1&0&0\\0&-1&0\\0&0&1 \end{pmatrix}$$ $$g2= \begin{pmatrix} 0&0&1\\ 1&0&0\\0&1&0 \end{pmatrix}$$ $$g3= \begin{pmatrix} .5&-.5\tau&.5/\tau \\ .5\tau& .5/\tau&-.5 \\ .5/\tau&.5&.5\tau\end{pmatrix},$$ with the spatial reflection being $$g4=\begin{pmatrix} -1&0&0\\0&-1&0\\0&0&-1\end{pmatrix},$$ where $\tau$ is the Golden Ratio.

Answer 2

The explicit matrix representations of all 5 irreducible representations of $I$ are given in Hu, Yong, Zhao, and Shu, "The irreducible representation matrices of the icosahedral point groups I and Ih", Superlattices and Microstructures, Volume 3, Issue 4, 1987, pages 391-398, DOI:10.1016/0749-6036(87)90212-6. You can construct the irreps of $I_h$ via the direct product structure with a point inversion.