I am dealing with the decomposition of the representation $5\otimes5$ of $SU(5)$:

$$5\otimes5=15\oplus10 $$

demonstration:

$$u^iv^j=\frac{1}{2}(u^iv^j+u^jv^i)+\frac{1}{2}(u^iv^j-u^jv^i)=$$

$$=\frac{1}{2}(u^iv^j+u^jv^i)+\frac{1}{2}\epsilon^{ijxyk}\epsilon_{xyklm}u^lv^m$$

where the term $\frac{1}{2}(u^iv^j+u^jv^i)$ has 15 independent components and the other has 10 components.

My question is: being the $\epsilon^{ijxyk}$ invariant in $SU(5)$, shouldn't the tensor $\epsilon_{xyklm}u^lv^m$ transform under the $\overline{10}$ representation, having 3 low free index?

(according to my notation an upper index transform under the $D$ representation while a lower index transforms under the $\overline{D}$ representation).

This post imported from StackExchange Physics at 2015-03-12 12:20 (UTC), posted by SE-user Caos