Understanding Electron Energy Splittings: The Role of Group Theory in Physics

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does representation theory help understanding electron energy splittings? if yes how? if no what is the role group theory plays in physics?
 
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Look at Tinkham's book on Group Theory and Quantum Mechanics, it is explained very well in there.

In general, symmetry principles play a huge role in physics, they simplify calculations.
 
Symmetry->representation->irreducible representation
 

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