Notation abuse warning: I will use the E7 series irrep names. You'll soon see why. 


In the general Lie algebra, the irrep you "start" with is $J$, the
adjoint. From the series for $J\bigotimes{J}=1+J+A...$ ($1$ is the 1-dimensional irrep) you get $A$, the antisymmetric one. (The quantum dimensions up to here are all given e.g. in Westbury's "R-matrices and the magic square" paper.) The next step would be $J\bigotimes{A}=J+A+S...$ where you can "fish out" the symmetric irrep $S$ from the Clebsch Gordan series. (Does anybody have the quantum dimension $Dim(S)=S(\alpha,\beta,\gamma)$ handy, just as for $J$ and $A$? In principle I could compute it myself, but this would speed up my work considerably.)  

Now comes the silly part :-) In the E7 series, $V\bigotimes{V}=1+J+S+A$ and I can now backwards compute the dimension of the defining irrep $V$ and see what comes out for, say, E8. Well, nonsense is coming out, which is hardly surprising. Still, I would have expected
integer nonsense or even better a zero ($324-273-52+1=0$ for F4 -the minus signs are only very mild cheating. :-)! Irrational nonsense is baaaad.  

So: Is it possible to bring all irreps, even from different rows of the magic square, in an 1:1 correspondence (if neccessary by inventing fictive irreps which still have integer dimension) or am I completely barking up the wrong Lie? 



"Fictive" irreps of the enveloping general Lie algebra
The (quantum) dimensions are all given in Vogel's papers (all but one unpublished). There may be some errors in my paper you refer to as I confused Vogel's parameters with the values of the quadratic Casimir.


I don't follow your "silly part". The representation $V$ is not defined for $E\_8$ and I don't see how you get a proposed quantum dimension.


