I am dealing with the decomposition of the representation 5⊗5 of SU(5):
5⊗5=15⊕10
demonstration:
uivj=12(uivj+ujvi)+12(uivj−ujvi)=
=12(uivj+ujvi)+12ϵijxykϵxyklmulvm
where the term 12(uivj+ujvi) has 15 independent components and the other has 10 components.
My question is: being the ϵijxyk invariant in SU(5), shouldn't the tensor ϵxyklmulvm transform under the ¯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 ¯D representation).
This post imported from StackExchange Physics at 2015-03-12 12:20 (UTC), posted by SE-user Caos