Integrity bases for tensors of type (GAMMA)(,r) whose components are polynomials in the components of tensors of type (GAMMA)(,5) ((GAMMA)(,6) for ('(d))O) are given explicitely for the double tetrahedral and octahedral point groups (('(d))T and ('(d))O), where the main axis of symmetry is trigonal. We formulate analytic basis states for the decomposition of SU(2) through the chain ('(d))T (R-HOOK) ('(d))C(,3) (R-HOOK) ('(d))C(,1) and use them to construct the Racah algebra. / A method is given for deriving branching rules, in the form of generating functions, for the decomposition of representations of SU(3) into representations of its finite subgroups. Interpreted in terms of an integrity basis, the generating functions define analytic polynomial basis states for SU(3) which respect the finite subgroup.

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