$SU(3)$ Tensor Methods in a Tetraquark 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. 
 A: Apologies for evincing magisterial cluelessness about what your diagrams represent and what you'd want to achieve, but I'd array the standard facts on tetraquarks avoiding Young diagrams, although they are self evident in the Dynkin labelling, which I also give, next to the tensor labelling. They may be useful to what you appear to be after--but I can't tell. 
$$
{\bf 3}\otimes {\bf 3}\otimes {\bf \bar{3}}\otimes {\bf \bar{3}} = 
(2) {\bf 1} \oplus (4) {\bf  8} \oplus {\bf 10} \oplus {\bf \overline{10}}
 \oplus {\bf 27} ~.
$$
Here,  the parentheses in front of the rep label denote its multiplicity in the C-G reduction.
$$D(1,0)={\bf 3}=\xi_j, ~  D(0,1)={\bf \bar{3}}=\xi^j;  \qquad  D(2,0)={\bf 6}=\xi_{jk},~  D(0,2)={\bf \bar{6}}=\xi^{jk}~;
$$
$$D(3,0)={\bf 10}=\xi_{jkl},~   D(0,3)={\bf \bar{10}}=\xi^{jkl} ; \qquad
  D(1,1)={\bf 8}=\xi_j^k, ~D(2,2)={\bf 27}=\xi_{jk}^{lm}.$$
(I do not have, nor would I waste any time on the text you seem to be trapped in. Have you tried Iachello? Cahn? Lichtenberg? Wu-Ki Tung? Wybourne? Carruthers? Gourdin? Coleman's "Aspects of symmetry"?)
Edit: In any case, from the Fierz identity of the Gell-Mann matrices I mentioned in my comment below (Okun's appendix; a straightforward consequence of their completeness when supplemented by the identity), the dot representing summation over the 8 adjoint indices a,
$$
{\bf \lambda}^k _i \cdot  {\bf \lambda}^l _j~ (w^i v^j+w^j v^i) =\frac{4}{3} (w^k v^l+w^l v^k)~ ,
$$
and analogously for the antisymmetrization of w and v, mutatis mutandis:
$$
{\bf \lambda}^k _i \cdot  {\bf \lambda}^l _j~ (w^i v^j-w^j v^i) =-\frac{8}{3} (w^k v^l-w^l v^k) ~ .
$$
