Symmetric three-spin-one subspace
= Symmetric three-spin-one subspace
The completely symmetric subspace of three spin-one systems has dimension
$$
\binom{3+3-1}{3}=10
$$
and decomposes into one total-spin-$3$ multiplet and one total-spin-$1$ multiplet:
$$
\operatorname{Sym}^3(1)=3\oplus1.
$$
The highest-weight product state generates the seven-dimensional spin-$3$ multiplet; its three-dimensional invariant complement is spin $1$.