⁷Many, many chemistry and physics texts make the unsupported claim that the spaces of degenerate eigenvecotrs of the hamiltonian always form irreducible representations. This is, in general, not the case. They provide no satisfactory mathematical proof of this. However, I have never run into a scenario where the eigenvectors are not irreducible. The only case in which it could really happen is that of a ”cosmic coincidence” - in which two wholly independent degenerate sets of eigenvectors happen to have the same energy as each other, and thus form a large space of degenerate eigenvectors. This happens sufficiently never in real physics to safely discount it; still, it’s important that it be acknowledged. This document also shows that it’s not necessary to make the assumption in the first place, as the process of finding the eigenvectors is unchanged.