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Optical selection rules in topological insulators Bi$_2$Sb$_3$, Bi$_2$Se$_3$, Bi$_2$Te$_3$ and Sb$_2$Te$_3$

We performed group theoretical investigation of symmetries of excitations in topological insulators \ce{Bi2Sb3}, \ce{Bi2Te3}, \ce{Bi2Se3} and \ce{Sb2Te3}, focusing on selection rules for optical processes. Electronic transitions of bulk states to bulk states, surface states to surface states and bulk states to surface states are studied over the entire Brillouin zone. A new technique is used to deal with transitions between surface states and bulk states. Time reversal symmtry is also included in the analysis. Our results show that only (Γ), (Λ) and (M) points of the Brillouin zone would depend on light polarization for transitions between bulk states and surface states. As an example of application, electron spin polarizations of photoelectrons are calculated at (Γ) point. A general rule for the direct product between the representation at an arbitrary point in the Brillouin zone with itself is presented.