Explanation of the Selection rule in light of IR spectroscopy I am in mathematics, but I am writing a paper on molecular vibrations (because you can use representation theory to handle this).

I want to prove the selection-rule, but I also have to give a presentation of my intermediate results. Can someone give a little (maybe trivial) explanation (in one or maximal two small sentences) of what the selection-rule does? (for example for the IR-spectrum). This is for my presentation if someone doesn't know something about the mathematical and chemical background.
The operator for interaction with infrared light is the permanent dipole of the molecule, $\mu$.  I don't know anything about representation theory, but in group theory, you decompose the molecule by its symmetry elements to find the eigenmodes.  The infrared active modes are those modes in which the permanent dipole changes over the course of a vibration.  An excellent reference is Symmetry and Spectroscopy by Harris and Bertolucci which should be available in your library.