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.



Can any Laurent polynomial symmetric under q->1/q and whose roots are all roots of unity be written as a ratio of products of quantum numbers?
Any polynomial with integer coefficients whose roots all are roots of unity are products of [cyclotomic polynomials](http://en.wikipedia.org/wiki/Cyclotomic_polynomial). Cyclotomic polynomials are in turn ratios of quantum integers.


For quantum groups you should be able to use a quantum analogue of the [Weyl dimension formula](http://en.wikipedia.org/wiki/Weyl_character_formula#Weyl_dimension_formula) to get explicit formulas.


