Editing Wigner–Eckart theorem (section)

===In terms of representation theory===

To state these observations more precisely and to prove them, it helps to invoke the mathematics of [[representation theory]]. For example, the set of all possible 4d orbitals (i.e., the 5 states ''m'' = −2, −1, 0, 1, 2 and their [[quantum superposition]]s) form a 5-dimensional abstract [[vector space]]. Rotating the system transforms these states into each other, so this is an example of a "group representation", in this case, the 5-dimensional [[irreducible representation]] ("irrep") of the rotation group [[angular momentum operator#SU(2), SO(3), and 360° rotations|SU(2) or SO(3)]], also called the "spin-2 representation". Similarly, the 2p quantum states form a 3-dimensional irrep (called "spin-1"), and the components of the position operator also form the 3-dimensional "spin-1" irrep.

Now consider the matrix elements <math>\langle 2p,m_1 | r_i | 4d,m_2 \rangle</math>. It turns out that these are transformed by rotations according to the [[tensor product]] of those three representations, i.e. the spin-1 representation of the 2p orbitals, the spin-1 representation of the components of '''r''', and the spin-2 representation of the 4d orbitals. This direct product, a 45-dimensional representation of SU(2), is ''not'' an [[irreducible representation]], instead it is the [[direct sum]] of a spin-4 representation, two spin-3 representations, three spin-2 representations, two spin-1 representations, and a spin-0 (i.e. trivial) representation. The nonzero matrix elements can only come from the spin-0 subspace. The Wigner–Eckart theorem works because the direct product decomposition contains one and only one spin-0 subspace, which implies that all the matrix elements are determined by a single scale factor.

Apart from the overall scale factor, calculating the matrix element <math>\langle 2p,m_1 | r_i | 4d,m_2 \rangle</math> is equivalent to calculating the [[projection (linear algebra)|projection]] of the corresponding abstract vector (in 45-dimensional space) onto the spin-0 subspace. The results of this calculation are the [[Clebsch–Gordan coefficient]]s. The key qualitative aspect of the Clebsch–Gordan decomposition that makes the argument work is that in the decomposition of the tensor product of two irreducible representations, each irreducible representation occurs only once. This allows [[Schur's lemma]] to be used.<ref>{{harvnb|Hall|2015}} Appendix C.</ref>