I am trying to understand the Georgi chapter of tensor methods in $SU(3)$ representations, and I don't know how to resolve the tensor product of 2 matrices in a 2 heavy quark + 2 light antiquark system. Namely:
Let the tetraquark system be: $QQ\bar q\bar{q}$
So, those $Q$ heavy quarks interact and can be understood in the $3\bigotimes 3$ representations for $SU(3)$, and $\bar q\bar q$ can be understood as the $\bar 3\bigotimes\bar 3=3\bigoplus\bar 6$ representations for $SU(3)$.
The interaction diagram for the $Q$ heavy quarks is shown in the following scheme, which is an analogy to the Feynman Diagram where a gluon is understood to be exchanged between upper and lower arrows:
$i'\xrightarrow{(T_{a})^{i'}_{i}}i , {}^iQ$
$j'\xrightarrow{(T_{a})^{j'}_{j}}j , {}^jQ$
So, as usuall we consider the elements of $3$ representation as $u^i$, and therefore the elements of $\bar 3$ representation as $v_{j}$.
I must understand how does the following expression transform:
$(T_{a})^{i'}_{i}(T_{a})^{j'}_{j}\cdot(1/2)\cdot (w^iv^j+w^jv^i)=\Xi\cdot(1/2)\cdot(w^{i'}v^{j'}+w^{j'}v^{i'})$
In other words, how can one give the correct value of $\Xi$?
Note1: $T_a$ is understood to be the $a-ith$ generator of the associated Lie Algebra, which come from the definition of the Gell-Mann Matrices.
$(T_{a})^{i}_{j}\propto(\lambda)_{ij}$, in matrix notation.
Note2: I understand the Young Tableaux and they are not intended to be used here, so please don't try to explain this situation through the tables.
Note3: I assume this is a purely solvable mathematical problem and doesn't need a deep insight into the physical concepts...
Also requesting information about how to understand the tensor products in these cases, since I have been checking many sources but the definitions seem not clear enough.