Question

In my hunt for spurious "alternatives" to the $E_7$ family I always encounter "fake" solutions. They turn out to be mostly $E_7$ family solutions disguised by $q\rightarrow{i*q}$. The effect is that the dimension (at $q=1$) comes out totally wrong. Generalizing a bit:
Take a random quantum dimension, say $d(G_2)=1+q^2+1/q^2+q^8+1/q^8+q^{10}+1/q^{10}$ and plug in a random root-of-unity $q=(-1)^{m/n}$. (Since even I know that the "interesting" q are these.) For small $n$ almost all values of $d$ will be integer.
1. This surely has to do with the fact that $d(q)$ is a ratio of quantum integers, i.e. cyclotomic?
2. Do these pseudo-dimensions have some "intuitive" meaning?

Answer 1

1. One explanation: for small values of n one gets integer quantum dimensions because the fusion category associated with small n is either degenerate or at least has very few simple objects. Getting q-dimension 0 means the weight is on a reflection of the outer wall of the Weyl alcove, so this is pretty common when n is small. Getting $\pm 1$ probably means that the weight is in the orbit of 0 under the shifted action of the affine Weyl group. Getting a negative number of any kind suggests that your weight is not in the Weyl alcove, but is a reflection of something that is. For type $G$ you will also encounter different behavior depending on if n is divisible by 3 or not.
2. The q-dimensions don't make proper sense unless the corresponding weight is in the Weyl alcove. When they are, they measure the growth rate of $End_{\mathcal{C}}(V^{\otimes n})$ where $V$ is the associated object in the fusion category.