I am little bit confused about the term anti-symmetric tensor. $$p_{ijk\ell}$$ is an anti-symmetric tensor. I would like to know its value when any of the two indices are same. For example, Is $$p_{1123}=0?$$

Also, do you have a good reference to cite these properties?


1 Answer 1


A totally anti-symmetric tensor means that exchanging any two indices gives rise to a minus sign. So suppose we have the case of two identical indices:

$$p_{iijk} = -p_{iijk}$$

where we exchanged the pair of $i$'s. Obviously if something is equivalent to negative itself, it is zero, so for any repeated index value, the element is zero.

There is also the case of an anti-symmetric tensor that is only anti-symmetric in specified pairs of indices. For example, one could declare $T_{abcd}$ is anti-symmetric only in the first two indices and symmetric in the other pair. This would mean,

$$T_{abcd} = -T_{bacd}$$

and that $T_{abcd} = T_{abdc}$. In terms of a reference to "cite these properties," if you mean for a paper or other academic work, this is considered common knowledge like say, the cross product, so there is no need to cite a text. However, you should find this topic treated in any book on tensor calculus, or a text on tensors in the context of the representation theory of finite groups, since this can be more abstractly described in terms of the action of the symmetric group.


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