The irreps of $H = SU(2) \times SU(2)$ are of the form $V_{j_1} \otimes V_{j_2}$ where $V_j$ is the spin-$j$ irrep of $SU(2)$ of dimension $2j+1$ (e.g., $V_0$ is trivial, $V_{1/2}$ is fundamental, and $V_1$ is adjoint). What are the irreps of $G = SU(2) \wr S_2 = (SU(2) \times SU(2)) \rtimes S_2$ where $S_2$ is the symmetric group on two letters?
When $j_1 \neq j_2$ the $S_2$ acts by swapping $j_1$ and $j_2$ and so I am guessing these irreps of $H$ lift to an irrep of $G$ like $(V_{j_1} \otimes V_{j_2}) \oplus (V_{j_2} \otimes V_{j_1})$. Is this correct?
But if $j_1 = j_2 = j$ then I am not sure what happens since $V_j\otimes V_j$ seems to be closed under the swap? Is $V_j \otimes V_j$ then itself already an irrep of $G$?
