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Have you ever met this kind of tensor expression?

[tex] A_{[a} C_{c]} [/tex]

That is, indices [tex] a, b[/tex] are anti-symmetric, and indices [tex] b, c[/tex] are anti-symmetric as well. I am confused by this, should I think this expression as: I anti-symmetrise indices [tex] a, b[/tex] first, and then antisymmetrise indices [tex] b, c[/tex]? this would result in

[tex] \frac{1}{4} (A_{a b} C_{c} - A_{b a} C_c - A_{a c} C_{b} + A_{c a} C_{b}) [/tex]

But, if I think of this expression by the meaning that I would get a minus sign whenever I exchange [tex] a, b[/tex], as well as I exchange [tex] b,c [/tex]. In this way, what I get should be

[tex] A_{[ab} C_{c]} [/tex]

So, which one is correct? I'm really confused...

Thanks for your help!

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