Descent of an SU(2) representation to SO(3)

A SU(2) representation descends to the SO(3) group exactly when $-I$ acts trivially. In ${\mathrm{Sym}}^n({\mathbb{C}}^2)$ it acts by $(-1{)}^n$, so descent holds precisely for even $n$, or integer spin $j=n/2$. This is the inflation of a group representation criterion for the kernel $\{\pm I\}$. Half-integer-spin representations instead remain representations of the covering group.