Wigner-Eckart Theorem and Simple Lie Groups

For simple Lie groups, matrix elements of vector operators (i.e. operators which transform according to the adjoint representation of the group) within an irreducible (finite-dimensional) representation, are studied. By use of the Wigner-Eckart theorem, they are shown to be linear combinations of the matrix elements of a finite number of operators. The number of linearly independent terms is calculated and shown to be at most equal to the rank l of the group. l vector operators are constructed explicitly, among which a basis can be chosen for this decomposition. Okubo's mass formula arises as a consequence.