Let $|s m\rangle$ denote the combined spin eigenstates for a system of two particles, each with spin 1. Derive expressions for all states with $m=s$ in terms of product states.

Given that the particles are identical, and that the spatial wavefunction describing their relative position has definite orbital angular momentum $\ell$, show that $\ell+s$ must be even. Suppose that this two-particle state is known to arise from the decay of a single particle, $X$, also of spin 1. Assuming that total angular momentum and parity are conserved in this process, find the values of $\ell$ and $s$ that are allowed, depending on whether the intrinsic parity of $X$ is even or odd.

[You may set $\hbar=1$ and use $J_{\pm}|j m\rangle=\sqrt{(j \mp m)(j \pm m+1)}|j m \pm 1\rangle .$ ]