Descent of an SU(2) representation to SO(3)
= 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 $\operatorname{Sym}^n(\mathbb C^2)$ 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.