Tensor index notation with e.g. square brackets I want to learn playing with indices and some notation in General relativity. But in every book just is used this notation. I know upper and lower but I don"t know the meaning of some combination of these indices, like when we have 3 indices in both side of the letter in bracket (e.g of the form $\partial_{[a}F_{bc]}$).
Could you please explain me or suggest a book that say almost everything , for example, when we need to put 2 indices in a tensor, one up and one of them down, ...?
 A: Interpret indices $[ijk]$ as a "determinant",
\begin{eqnarray}
    [ijk] \rightarrow
    \left| \begin{matrix} i & j & k \\ i & j & k \\ i & j & k \\ \end{matrix} \right|.
\end{eqnarray}
Expand above indices,
\begin{eqnarray}
    [ijk] \rightarrow
    ijk + jki + kij - ikj - jik -kji.
\end{eqnarray}
Set them to a tensor $T_{[ijk]}$ (including signs) and divide by normalization factor, so we get
\begin{eqnarray}
   T_{ [ijk] } =
   \frac{1}{3!} \left(T_{ijk} + T_{jki} + T_{kij} - T_{ikj} - T_{jik} - T_{kji} \right).
\end{eqnarray}
I think it can be done because of antisymmetrization structure of a determinant.
A: Here's various things used in index notation : 


*

*Types of indices : 


*

*Greek indices for spacetime indices (tensor indices)

*Lower-case latin indices for 


*

*spacelike components 

*local Lorentz components 

*group components (for gauge indexes)


*Upper case latin indexes for spinor indexes

*Dotted upper case latin indices for conjugate spinor indexes. Generally speaking, dotted indexes indicate complex conjugate indexes. 


*Upper index : Denotes the components of a vector, or a basis of dual vectors. Example : $V^\mu$ is the components of a vector, $dx^\mu$ is a basis of dual vectors, $T_\mu dx^\mu$ is a dual vector.

*Lower index : Denotes the components of a dual vector, or a basis of vectors. Example : $T_\mu$ is the components of a dual vector, $\partial x_\mu$ is a basis of vectors, $V^\mu \partial x_\mu$ is a vector. 

*Parenthesis : Denotes the symmetrization of a tensor with respect to those indices. That is, for $n$ indices, $$T_{(\alpha\beta\gamma...)} = \frac{1}{n!}\sum_{p \in \mathrm{permutations}} T_{p(\alpha\beta\gamma...)}.$$ Examples : $T_{(\alpha\beta)} = \frac{1}{2} (T_{\alpha\beta} + T_{\beta\gamma})$, $p^{(i}q^{j)}=\frac12(p^iq^j+p^jq^i)$. Note, in particular, that the parenthesis notation can span indices over multiple tensors.

*Brackets : Denotes the antisymmetrization of a tensor with respect to those indices. That is, for $n$ indices, $$T_{[\alpha\beta\gamma...]} = \frac{1}{n!}\sum_{p \in \mathrm{permutations}}(-1)^{n_p} T_{p(\alpha\beta\gamma...)} $$ where $n_p$ indicates the number of single permutations of $p$. Example : $$T_{[\alpha\beta\gamma]} = \frac{1}{6} (T_{\alpha\beta\gamma} - T_{\gamma\beta\alpha} + T_{\beta\gamma\alpha} - T_{\alpha\gamma\beta} + T_{\gamma\alpha\beta} - T_{\beta\alpha\gamma})$$
Like the parenthesis notation, it can span over multiple tensors. Example : $f_{[\alpha}g_{\beta]}=\frac12(f_\alpha g_\beta - f_\beta g_\alpha)$

*Comma : Denotes the partial derivative with respect to this component. Example : ${V^\alpha}_{,\beta} = \partial_\beta V^\alpha$

*Semicolon : Denotes the covariant derivative with respect to this component. Example : ${V^\alpha}_{;\beta} = \partial_\beta V^\alpha + {\Gamma^{\alpha}}_{\beta\gamma} V^\gamma$


Less commonly used : 


*

*Pipe : Denotes the covariant derivative with respect to local Lorentz indices. Example : ${V^a}_{\vert i} = \partial_i V^a + {\omega^a}_{bi} V^b$

*Colon : Denotes the covariant derivative with the Levi-Civita connection. Example : ${V^\alpha}_{:\beta} = \sum_{\alpha,\beta,\gamma} (\partial_\alpha V^\alpha + \{^\alpha_{\beta\gamma}\} V^\gamma)$

