Quantum dimensions are quantum integer fractions (or so I heard). Example: $G_2(\lambda_2)$ (technically it should read q^{1/6}, I know...)

$q^{-10}+q^{-8}+q^{-2}+1+q^2+q^8+q^{10}$ = $q_{12}*q_7*q_2/q_6/q_4$

where $q_n$ is shorthand for $(q^n-q^{-n})/(q-1/q)$.

Obviously the left hand side must be symmetric under q->1/q and all roots
are roots of unity. Are these conditions already sufficient that a given
Laurent polynome can be converted into a quantum integer fraction? Is there
even a constructive algorithm? (BTW, I believe the result should be unique
since each quantum integer $q_n$ introduces a new root of unity.)
In the example the roots are $(-1)^{m/7}$ and $(-1)^{m/12}$ so I might
begin with $l.h.s./q_{12}/q_7$, compute the roots of that etc., and this
algorithm *might* already work.