It's common knowledge among physicists that when you have a linear Hermitian operator like a quantum Hamiltonian that commutes with symmetry matrices you can then block-diagonalize Hermitian operator into sectors that have different eigenvalues of the symmetry matrix. This has all sorts of implications for forbidden transitions, topological indices, etc. In fact it's so well-known I'm not sure what reference to use for it. Does anyone know of a good review article or book that discusses how discrete symmetries are used in this way? Ideally it would go into some advanced detail, discussing applications and less common cases, like antisymmetries.
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$\begingroup$ wouldn’t that be covered in any group theory test that includes finite groups? $\endgroup$– ZeroTheHeroCommented Jun 25, 2021 at 20:36
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$\begingroup$ ZeroTheHero: You might think so, but is it? For example, I don't see anything about block-diagonalization in Dresselhaus's "Applications of Group Theory to the Physics of Solids". And really, group theory is overkill for this, I just want to show that if $S^\dagger H S = H$ then the element of $H$ between two states with different symmetries under $S$ must vanish. $\endgroup$– DavidCommented Jun 25, 2021 at 21:11
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