I am working out the exchange symmetry of the eigenstates of the total angular momentum operator of a system of two spin-1 bosons.
I know that there must be a quintet, triplet, and a singlet state.
The highest state of the quintet is $|\uparrow\rangle |\uparrow\rangle$ and is symmetric under particle exchange.
Applying the lowering operator in this state generates the entire quintet. Since the lowering operator is symmetric itself, all states in the quintet are symmetric.
Using the orthogonality of states, one can deduce the highest state of the triplet. Successively applying the lowering operator generates the entire triplet.
My problem is: How does one show that the triplet states are antisymmetric, given that the quintet states are symmetric.
Phrased more generally perhaps, if the state $|\Psi\rangle $ is symmetric under particle exchange and state $|\Phi\rangle$ is such that $\langle \Psi | \Phi \rangle=0$, can one show that $|\Phi\rangle$ is antisymmetric?