Wigner coefficients for a semisimple Lie group and the matrix elements of the O(n) generators

A purely algebraic infinitesimal method for obtaining multiplicity-free Wigner coefficients is presented. The method is applied to obtain analytic expressions for the complete matrix elements of all O(n) generators. Moreover the structure of these matrix elements in terms of reduced matrix elements, Wigner coefficients, and reduced Wigner coefficients is made explicit. By comparison with the Wigner–Eckart theorem explicit analytic expressions are obtained for the fundamental Wigner coefficients of O(n). Finally the results are presented in a form which is directly analogous with the corresponding results for U(n).