Two identical spin-one bosons

For two spin-one particles, the spin tensor product has a six-dimensional symmetric subspace and a three-dimensional antisymmetric subspace. A symmetric spatial state combines with symmetric spin, while an antisymmetric spatial state combines with antisymmetric spin, so that the total bosonic state remains symmetric.

If $E_n=E_0+n\Delta$, the two-particle level $2E_0+N\Delta$ has degeneracyEach unequal orbital pair contributes six symmetric-spin states and three antisymmetric-spin states; an equal orbital pair exists only for even $N$ and contributes six states.