How to Structure a Degenerate Space

Theorem 1 If a matrix commutes with all the matrices of an irreducible representation, the matrix must be a multiple of the unit matrix  

In octahedral transition-metal complexes, the i-orbitals, which transform as the E irrep, are and d 2 y2. They are seen to match E9) and Ee), respectively. The 

This operator connects the two components, and, therefore, if we require that the matrix be of the specified form, then the relative phase freedom is lifted. With such a connecting element, one can indeed easily construct a proper ladder operator  

In Appendix D we list standard conventions that are frequently used to define canonical basis sets for degenerate irreps. 

For the octahedral T state, the standard basis complies with the transformation 

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