Understanding Anti-Symmetric Tensor Expression

[ismaili]

Dear guys,

Have you ever met this kind of tensor expression?

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

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

$$\frac{1}{4} (A_{a b} C_{c} - A_{b a} C_c - A_{a c} C_{b} + A_{c a} C_{b})$$

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

$$A_{[ab} C_{c]}$$

So, which one is correct? I'm really confused...
Thanks for your help!

 

[xepma]

Probably they mean that you antisymmetrize over the indices a and c only, while the index b is left unchanged.

 

[ismaili]

Probably they mean that you antisymmetrize over the indices a and c only, while the index b is left unchanged.

Thanks for your discussion.
But, no, in that case the expression is written as

$$A_{[a|b|} C_{c]}$$

where the index $$b$$ is enclosed by two bars, $$| |$$.