Understanding Anti-Symmetric Tensor Expression

[ismaili]

Dear guys,

Have you ever met this kind of tensor expression?

A_{[a <b>} C_{c]} </b>

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, | |.