Variants of this question have been asked before, and techniques for deriving all 8 eigenstates of the total system have been provided, most clearly in this post: Adding 3 electron spins
My question is about what arguments can be made if one is given a series of states, such as,
$$\frac{1}{\sqrt{2}}(|↑↓↑⟩-|↓↑↑⟩)$$ $$\frac{1}{\sqrt{3}}(|↑↑↓⟩+e^{2i\pi/3}|↑↓↑⟩+e^{-2i\pi/3}|↓↑↑⟩)$$
and we ask whether or not these are eigenstates of spin angular momentum for the total system. I can think of necessary conditions, like that these are orthogonal to the total system eigenstates I know (the easy ones $|↑↑↑⟩$ and $|↓↓↓⟩$). But this is of course not a sufficient condition. I can act with the lowering or raising operators on these provided states, and get other states, but that doesn't tell me that any of these states are eigenstates of the total system.
I would think that there's a straightforward argument as to why the first of the given states isn't an eigenstate of total spin, since it's a superposition of only two of the possible $s_z^\mathrm{total} = \frac{1}{2}$ states. But I haven't been able to construct that coherently. What could be a general approach here, or do you need to construct the states explicitly?
