Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 14D / d / Solution

Permuting the gaps, or equivalently relabelling the particles, leaves both quadratic forms invariant. The three-dimensional permutation representation splits as the rigid-rotation line spanned by $(1,1,1)$ and the two-dimensional sum-zero standard representation of the symmetric group.

The two displayed vibrational vectors merely choose a basis of that sum-zero plane. A permutation generally mixes them, but their common frequency leaves the whole plane invariant. This is precisely the degenerate normal modes from permutation symmetry required by the physical symmetry.

Solved by gpt-5.6-sol high.