As a chemist, I do this all the time...for symmetry groups. Which are finite, luckily :-) For knot theory purposes, I'd like to have a complete list of Lie group irreps R with the property that the tensor product $R\otimes{R}$ (or is that $R\otimes{adj(R)}$??)=$1\oplus{R'}\oplus{R''}...$ has at most 8 summands.
Luckily, Hayashi (J. Phys. A: Math. Gen. 27 (1994) 2407) can be plundered, and I assume that the cases given there are already complete and higher irreps give rise to far more than 8 summands for any of the ABCDEFG Lie groups. The <=8 list extracted from there is R=
E8(248),E7(56),E7(133),E6(27),E6(78),F4(26),F4(52),G2(7),G2(14),BCD($\lambda_2$),BCD($2\lambda_1$) and for the A series I am confused as I don't know when it's $R\otimes{R}$ and when $R\otimes{adj(R)}$ where the "1" pops up. So I'd like to ask you if this list is really complete. I expect 10 summands for BCD($\lambda_3$), BCD($\lambda_2+\lambda_1$)..., tons for EFG (E6(351) and E6(351') may stay below my limit), and A($\lambda$)*A(adj($\lambda$)) might also add a few examples (if I recall correctly, A2(8)*A2(8) has 6 terms). I read Fulton&Harris but didn't understand that much, otherwise I wouldn't pester you :-)
(Conjecture: The summand number+1 in $R\otimes{R}$ = the number of linear independent tangles needed for a skein relation for the Reshetikhine-Turaev invariant based on R. Proof: Found no counterexample yet :-))
(Note: Hayashi deals with quantum groups, but the summand number should be independent of any q-deformation, right?)