Sorry for this silly question, but this notation keeps popping up in my face in too many different places without any clarification, so I will be so grateful if someone can help and answer my question: Does the following notation
$\alpha_{[\mu_1...\mu_p}\beta_{\nu_1...\nu_q]}$
mean this
$\alpha_{[\mu_1...\mu_p]}\beta_{[\nu_1...\nu_q]}$
where the square brackets are the antisymmetrization brackets?
Please help, I need to know and it's driving me crazy!
No, it means that you should antisymmetrize the expression over all indices $\mu_1 \dots \mu_p \nu_1 \dots \nu_q$.
For example, $\alpha_{[\mu} \beta_{\nu]} = \frac12 (\alpha_\mu \beta_\nu - \alpha_\nu \beta_\mu)$ (or maybe without the factor $1/2$, depending on which convention you use).