I have a problem opening up an antisymmetric tensor. Please pardon my jargon I may not express it the best way.
I need to open up:
$T^{bk}_{\enspace \enspace [q}T^{df}_{\enspace \enspace r]}$
This I believe is a set of anti-symmetric tensors of order 4.
I am using this as reference:
$t_{[ab...c]}=\dfrac{1}{N!}$(alternating sum over all permutations of the indices $a,b, ...,c$)
and the book as an example gives an anti-symmetric tensor of order 3
$t_{[abc]} = \dfrac{1}{6}$($t_{abc}-t_{acb}+t_{cab}+t_{bca}-t_{bac}$).
Now my issue is this, when I open up my tensors how would I take 'an alternating sum over all permutations of the indices'. I have used a version of the inclusion-exclusion principle do you think that this is correct?
By inclusion-exclusion principle I mean that those indices which are cyclic i.e. $t_{abc}$, $t_{cab}$, so on get a $+$ve sum while the anti-cyclic indices get a $-$ve sign. Do you think that this is correct?
So my net result is:
$\dfrac{1}{4!} \left[ \sum(\text{permutations which are cyclic})-\sum(\text{permutations which are not cyclic}) \right]$
I am not sure about this result.
Thank you very much for your assistance.