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I'm reading the SU(N) chapter in Jones' Group theory book. In SU(3) we have these 3 component spinors which transform as [tex] \psi^{'}_{a}=U_{a}^{..b}\psi_{b} [/tex] and we have upper spinors defined by [tex]\psi^{a}=\epsilon^{abc}\phi_{[bc]}[/tex]

Now if consider building up higher-dimensional reps, by taking Kronecker products. Starting with [tex] 3 \otimes \bar{3} [/tex]:

[tex]\psi_{a}\bar{\psi}^{b}=\left( \psi_{a}\bar{\psi}^{b}-\tfrac{1}{3}\delta^{b}_{a}\psi_{c}\bar{\psi}}^{c}\right)+\tfrac{1}{3}\delta^{b}_{a}\psi_{c}\bar{\psi}^{c} [/tex]

Now I understand that the last term is invariant, which already suggests to me it's a singlet, but I can't see how this term is a completely antisymmetric combo of all three indices. Could anyone show me explicitley why this is?

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# Can anyone explain this term is antisymmetric in SU(3)

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