Reduction of quantum systems with symmetry, continuous and discrete

Reduction of dynamical systems is closely related with symmetry. The purpose of this article is to show that Fourier analysis both on compact Lie groups and on finite groups serves as a reduction procedure for quantum systems with symmetry on an equal footing. The reduction procedure is applied to systems of many identical particles lying in R 3 which admit the action of a rotation group SO(3) and of a symmetric or permutation group.