I would like to use rotation matrices as representations of the rotation group. I would like to know if these representations are faithful, i.e. isomorphic to the rotational group elements.
I read on the bottom of p. 61 in Ref. 1 that
"Only the $j = 1$ representation is isomorphic to the rotation group itself."
Can someone explain to my why this is the case?
Note: $j=1$ means that the eigenvalue of $J^2$ is $j(j+1)$, where $J^2=J_x^2+J_y^2+J_z^2$, where $J_i$ is the generator of rotation about the $i$-axis.
- J. Tseng, Symmetry and Relativity, lecture notes, 2017. The PDF file is available here.