I am recently study group theory and its application in quantum mechanics, but got stuck at a very important point that how group theory can be applied to analyze energy level degeneracy.
In many textbooks, the statement what I understand is like this: if you find all symmetry operations that leave Hamiltonian invariant (the group of schrodinger equation) and get all the irreducible representations of them, then the dimension of each irreducible representation corresponds to the degeneracy of an energy level, though the order of different energy level cannot be predicted.
It is proved in many textbooks that the wave functions of an energy level can form a basis, and the matrices representing the symmetry transformation among them are indeed representation of the group. But why is that an irreducible representation of the group? (in the case of finding all the symmetry operations, so no accidental degeneracy argument) Or in other words, can an energy level degeneracy correspond to addition of several irreducible representations?
For example, in Jorio's book 'Group theory, application to the physics of condensed matter', it mentions that what if a representation under eigenfunctions of an energy level can be broken into smaller irreducible representations. It says 'But if this happened, then the eigenvalues for the two subsets would be different, except for the rare case of accidental degeneracy.' But why different irreducible representations correspond to different energy eigenvalues?
It is different from the question 'What is the relationship between symmetry and degeneracy in quantum mechanics?' because the problem why cannot an energy level degeneracy correspond to addition of different irreducible representations (even in the case all symmetries have been found) is not answered.
