One final question, $G_2$ has the fusion rule ${\bf 7}\otimes{\bf 7}\rightarrow {\bf 1}_s + {\bf 28}_s+{\bf 7}_a+{\bf 14}_a$ so the trivalent vertex is antisymmetric not symmetric. Is this what you mean, or is there some clever way to fix this so that the vertex is symmetric?

Thanks for the reference! I'm confused though about $\mathsf{Rep}(S^n)$, as this seems to contradict Corollary 8.9 unless I've misunderstanding something.

For the special case where ${\bf a}\approx {\bf n}$ what are you able to say? I know the fundamental of $S^{n+1}$ works (and obviously so does $\mathbb Z_2 \times S^{n+1}$), are there other examples?

Thanks for pointing this out, I've fixed the location of the $\pi$s

It’s possible, a colleague did the numerics and he may have made a typo in the notes. Did you get the 1/y term analytically?