# Tensor question

christodouloum
I am a bit confused by this observation.
Every tensor is it's symmetric plus antisymmetric part.

Thus for the components of a (0,3) tensor

$$F_{\lambda\mu\nu}=F_{[\lambda\mu\nu]}+F_{\{\lambda\mu\nu\}}$$

and if I write this down explicitly I end up that for the components of ANY (0,3) tensor

$$F_{\lambda\mu\nu}=(1/3)(F_{\lambda\mu\nu} +F_{\mu\nu\lambda}+F_{\nu\lambda\mu} )$$

Huh? Does this indeed hold?

christodouloum
I am replying to my self since I searched around a bit and the statement
Every tensor is it's symmetric plus antisymmetric part

holds for every pair of indices not generally. So now I know that the last equality I wrote does not hold but still, is there any way to generalize this idea? I mean, please correct me if I am wrong but we have

$$F_{\lambda\mu}=F_{[\lambda\mu]}+F_{\{\lambda\mu\}}$$
$$F_{\lambda\mu\nu}=F_{[\lambda\mu]\nu}+F_{\{\lambda\mu\}\nu}$$

how about a relation between $$F_{\lambda\mu\nu}$$ ,$$F_{\{\lambda\mu\nu\}}$$ and $$F_{[\lambda\mu\nu]}$$??