I'd like to implement an algorithm which produces matrix representations of the (complexified) Lie Algebra $\mathfrak{su}(3)$ on carrier spaces with arbitrary highest weight vector; i.e. 8 $n\times n$ matrices $T_a$, $a=1,..,8$ with complex entries which satisfy $$[T_i,T_j]=i \sum_{k=1}^{8} c_{ijk}T_k $$ where $c_{ijk}$ are the $\mathfrak{su}(3)$ structure constants.
Let me first tell you what I have investigated so far:
1 - Tensor product
It is known that ($n>m$) $$(n,0)\otimes (0,m) \simeq (n,m) \oplus (n-1,m-1) \oplus \ldots \oplus (n-m,0)$$ where $(n,m)$ refers to the carrier space of the irreducible representation with highest weight $(n,m)$ (as usual).
The matrices acting on the left hand side are easily constructed via the Kroneckerproduct. The question is now, how can one extract the $(n,m)$ space from it. The only way I see is to get to know the above isomorphism explicitely (which would be the analog of knowing the Clebsch-Gordon coefficients in the $su(2)$ case) and this problem seems to be much more difficult to solve than my original one.
2 - Using $\mathfrak{su}(2)$-strings
Assuming one knows the connection of all weight vectors to the su(2)-strings one could calculate its ladder operators (the describing equations would result in a linear system in the matrix coefficients) and from it one could reconstruct the 8 su(3) matrix-reps.
The problem is finding these connections: I first thought I could reconstruct the su(2) strings with the same algorithm one finds all weights from the Cartan Matrix. But I run into trouble if weight vectors with multiplicites $>1$ are appearing. Even if I'd know the multiplicities I couldn't find a rule for: which string belongs to which weight vector.
3 - Using the commutation relations
I even tried to solve $$[T_i,T_j]=i \sum_{k=1}^{8} c_{ijk}T_k $$ directly. As Ansatz for the matrices I take general hermitian tracefree $n\times n$ matrices (which results in $n^2-1$ parameters).
The first problem is that this is a quadratic equation in the matrix coefficients and Mathematica seems to be terrible slow in solving such a system exactly. Furthermore the $n^2-1$ parameters allow additional freedom (for example choosing some matrices to be real) and it seems not practical for automation since one would have to fix the remaining parameters for each case separatly.
Any comments on why the approaches are good/bad and how to fix them or suggestions for a complete different approach are very welcome!