Let φ SU(2) ---> SO(3) be the double covering Any irreducible representation of SO(3) pulls back by φ to provide an irreducible representation of SU(2) on the same finite dimensional Hilbert space. this seems clear, almost not worth saying: the pullback is obviously irred. and has the same dimension. I have a question about the other direction----suppose an irred. rep. of SU(2) factors thru φ then it obviously gives a representation of SO(3), same space same dimension and all, but under what circumstances is the representation irreducible?