I am currently working my way through Howard Georgei's text on Lie Algebras in Particle Physics. I am having some trouble understanding how to go about decomposing a general tensor product into irreducible representations and counting the dimensions. One of the examples given is:
$$u^iv^j = \frac{1}{2}(u^iv^j + u^jv^i) + \frac{1}{2}\epsilon^{ijk}\epsilon_{klm}u^lv^m.$$
I understand that the first term on the right side transforms like a 6, as it is symmetric in the upper indices. I am having trouble understanding how the dimension for the second term has been counted. The textbook says that as this term has only 1 lower index, it transforms as $\bar{3}$. In doing so, we appear to ignore the $\epsilon^{ijk}$ term and only count the indices in the remaining. Why is this? I am also having trouble understanding how one uses this levi-civita to antisymmetrize in general.
For concreteness, a later example:
$$u^iv^j_k = \frac{1}{2}(u^iv^j_k + u^jv^i_k - \frac{1}{4}\delta^i_ku^lv^j_l - \frac{1}{2}\delta^j_ku^lv^i_l) + \frac{1}{4}\epsilon^{ijl}(\epsilon_{lmn}u^mv^n_k + \epsilon_{kmn}u^mv^n_l) + \frac{1}{8}(3\delta^i_ku^lv^j_l-\delta^j_ku^lv^i_l)$$
says that the right hand side is a $15\oplus\bar{6}\oplus 3$.
I would highly appreciate it if someone could explain what the three terms on the right hand side mean and how the counting is done.
