# Tensor product decomposion in Lie algebras

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?)

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Does your adj(R) mean the dual representation of R? Also, physicists are notorious for using the dimension as the name of an irreducible representation instead of the more precise highest weight (as you noticed). Anyway, the general formula for the decomposition of the tensor product of two simple modules as a direct sum of irreducible ones is well known. See e.g. Humphreys' book Introduction to Lie algebras and representation theory (Springer GTM series). These are in terms of (formal) characters. Some tables of those have been published (IIRC extensive enough to cover your cases). –  Jyrki Lahtonen Aug 19 '11 at 20:23
Hauke, sorry about writing the earlier bit in haste. The result that I had mind (that also helps you) is known as Steinberg's formula. It gives you the decomposition factors of any tensor product of two simple modules. It is a generalization of what you may know as the Clebsch-Gordan formula that says what happens when you add two (quantum mechanical) angular momenta together. –  Jyrki Lahtonen Aug 20 '11 at 19:36