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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.

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If coeffieicnets are in Z, then yes. Then q^N*Laurent poly divides q^M-1 so is a product of cyclotomic polynomials, hence can be written in the given form. –  Peter McNamara Aug 21 '11 at 20:59

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up vote 5 down vote accepted

Any polynomial with integer coefficients whose roots all are roots of unity are products of cyclotomic polynomials. 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 to get explicit formulas.

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