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.
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.