Yes, exactly. that's what I wrote in post #7.
After introduction of the normal coordinates, the qm problem reduces to that of a product of harmonic oscillator eigenstates. The ground state is a product of gaussian functions and can easily be seen to be totally symmetric. The states with one quantum correspond (up to a normalization constant) to one where the ground state function is multiplied by q_i, the coordinate of the ith normal coordinate which transforms according to one of the irreps you wrote down.