Past exam of the condensed matter physics major option course of the University of Oxford / 2016 / 2 / b

For the full prism point group, a threefold rotation eliminates any invariant basal component of a polar vector, while the horizontal mirror eliminates its axial component. Therefore the phase preserving all of $D_{3h}$ is nonpolar, despite lacking inversion. A polar phase would have to break some of these symmetries.

The indicated six vertex directions comprise three basal directions and their horizontal-mirror partners. Crucially, this orbit does not contain the negatives of those vectors. Choose $z$ along the prism and $x$ toward one basal vertex. The cubic polynomialis invariant under the prism's rotations and mirrors. Along a vertex direction $\mathbf{P}=P\hat{\mathbf{n}}$ with a nonzero basal component, it is proportional to $P^3$, so odd scalar terms are symmetry-allowed. Along the prism axis the horizontal mirror instead sends $P_z\mapsto -P_z$, forcing an even free energy. Merely omitting inversion is insufficient to decide either polarity or the allowed invariants.