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?

  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.

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.