It seems there should be a list of tensor identities on the internet that answers the following, but I can't find one.(adsbygoogle = window.adsbygoogle || []).push({});

For tensors in ##R^4##,

##S = S_\mu{}^\nu = S_{(\mu}{}^{\nu)}## is a symmetric tensor.

##A = A_{\nu\rho\sigma}= A_{[\nu\rho\sigma]}## is an antisymmetric tensor in all indices.

The basis vectors are suppressed for simplicity.

Parentheses ##(\ \ )## enclose symmetric indices; the tensor remains the same upon an exchange of any two indices.

Brackets ##[\ \ ]## enclose antisymmetric indices; the tensor changes sign on an exchange of any two indices.

Is the inner product ## SA=S_\mu{}^\nu A_{\nu\rho\sigma}## antisymmetric in all indices?

(In a related aside, ##A_{\nu\rho\sigma}## can be expressed as a scalar times a "unit valued" antiysmmetric tensor. Call it ##\epsilon^3{}_{\mu\nu\rho}## in ##R^4##, or the 3 index Levi-Civita tensor in 4 dimensions. But I don't see any internet reference to this animal either. Are my keywords lacking?)

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# Product of a symmetric and antisymmetric tensor

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