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Can somebody explain in a simple way why, talking about representations, $3\otimes3=3\oplus6$, $3\otimes\bar{3}=1\oplus8$ and $3\otimes3\otimes3=1\oplus8\oplus8\oplus10$?

Here $3$ and $\bar{3}$ are the fundamental and anti-fundamental of $SU(3)$, in this case.

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Related $SU(3)$ post: physics.stackexchange.com/q/10403/2451 especially the answer physics.stackexchange.com/a/14586/2451 . Do you know Clebsch-Gordan decomposition of $SU(2)$ irreps? See e.g. physics.stackexchange.com/q/16098/2451 . –  Qmechanic Mar 21 '13 at 17:29
I found the group theory appendix (B) in Zee, Quantum Field Theory in a Nutshell, to be helpful for this stuff. –  Michael Brown Mar 22 '13 at 0:19

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