Algebraic expressions of the Clebsch-Gordon coefficients of the point group T

A method for finding algebraic expressions of the Clebsch-Gordan (CG) coefficients of point groups is proposed and applied to the tetrahedral group. It is shown that in constructing the CG coefficients the irreducible symmetry operator (ISO) of a double group G dagger can be replaced by an effective ISO which is much simpler than the usual ISO. The effective ISO for the group chain T dagger superset of C(3)dagger is P(mu<(mu)over bar>)((lambda)) = delta(mu<(mu)over bar>) +3d(mu<(mu)over bar>)((lambda))(C-2z)*C-2z, where d((lambda)) (C-2z) is the matrix of C-2z in the irrep lambda of T dagger. With this effective ISO and the algebraic expression of d((lambda)) (C-2z), the algebraic expressions are derived for the real CG coefficients of T dagger in the group chain T dagger superset of C(3)dagger. The algebraic expressions for the complex (real) CG coefficients of the group chain T dagger superset of D(2)dagger superset of C(2)dagger (T dagger superset of C(2)dagger) have also been obtained. (C) 1998 American Institute of Physics. [S0022-2488(98)04110-3].