Three-quadrupole-phonon multiplet ($S^3(V_2)=V_0\oplus V_2\oplus V_3\oplus V_4\oplus V_6$)

Three identical quadrupole quanta occupy the third symmetric power of the five-dimensional spin-two space. Its dimension is $\left(\genfrac{}{}{0ex}{}{7}{3}\right)=35$, and its irreducible representations have $J=0,2,3,4,6$, each once. All have positive parity. This counts allowed angular states in the ideal harmonic nuclear quadrupole vibration; anharmonic interactions can split their energies.