I am wondering how to write rotations of binary icosahedral group as matrices. For example, the identity element of this group corresponds:
$R_1 = 1 + 0i + 0j + 0k$
since this group corresponds to spin3, it is a 3*3 identity matrix.
$R_2 = 0 + 1i + 0j + 0k$
where i,j,k can be thought as pauli_X, pauli_Y and pauli_Z matrices R2 corresponds $pi$ rotation about an edge aligned with an axis. How to write it as a matrix?
Same with R3:
$R_3 = 1/2 + 1/2i + 1/2j + 1/2k$
which is a $2pi/3$ rotation about one of the faces centered in an octant.