Spin-orbit coupling and double groups

The spin-orbit coupling term in the Hamiltonian induces the coupling of the orbital and spin angular momenta to give a total angular momentum J = L + S. This results in a splitting of the Russell-Saunders multiplets into their components, each of which is labeled by the appropriate value of the total angular momentum quantum number J. The character of the matrix representative (MR) of the operator R(0 n) in the coupled representation is 

A detailed analysis (Chapter 11) shows that this result depends upon the commutation relations for the L operators, and, since the spin and the total angular momentum operators obey the same commutation relations (CRs), this formula holds also for S and for J  

In configuration space we would expect a rotation through 0 + 2n to be equivalent to a rotation through 0. The curious behavior implied by eq. (5) arises because our state 

Exercise 8.1-1 Name the classes in the double point groups C4 and C2v. 

Rewriting eqs. (2) and (3) in a slightly more convenient notation, we have 

---