Question

I take a random but practical example direct from "R-matrices and the magic square" by Bruce Westbury: The adjoint irrep $A$ of the $E_7$ family has quantum dimension
$qi[2*m+3]*qi[3*m/2+2]*qi[3*m/2]/qi[m/2]/qi[m/2+2]$
where $qi[m]$ (at the value $q$) denotes the quantum integer function and $m$ is "mostly" an integer (but not throughout, e.g. $G_2$ has $m=-5/2$). At $q=1$, we get $(m+4)|120$. Let's stay at integer $m$ for simplicity and take, say, $m=116: Dim(A)=2068$. OK, now a different value, say, $q=i: Dim(A)=-44/5$. Owch.
I conclude that $qi[58]*qi[60]|qi[174]*qi[176]*qi[235]$ (in the Laurent polynomial sense, see below) fails to hold, although $58*60|174*176*235$.
Can you give an condition for $qi[m]|qi[n]$ (or even better, for arbitrary products of quantum integers in numerator and denominator allowed) that is sharper than $m|n$? $|$ shall mean that the denominator of the fully cancelled form for $qi[m]|qi[n]$ only may contain some $q^k$ (i.e. not $q-1/q$ or something like that). (Random Example: $qi[2]*qi[4]|qi[8]$ is false because the denominator contains $1+q^8$.)

Answer 1

Factor a fraction in the $q$-integers as a product of cyclotomic polynomials $\Phi_d$, using that $q^n-1 = \prod_{d|n} \Phi_d$, and look at the multiplicity of each polynomial $\Phi_d$.