An easily programmable algorithm is given for the computation of SO(5) spherical harmonics needed to complement the radial (beta) wave functions to form an orthonormal basis of wave functions for the five-dimensional harmonic oscillator. It is shown how these functions can be used to compute the (Clebsch–Gordan a.k.a. Wigner) coupling coefficients for combining pairs of irreps in this space to other irreps. This is of particular value for the construction of the matrices of Hamiltonians and transition operators that arise in applications of nuclear collective models. Tables of the most useful coupling coefficients are given in the Appendix.

This article reviews many manifestations and applications of dual representations of pairs of groups, primarily in atomic and nuclear physics. Examples are given to show how such paired representations are powerful aids in understanding the dynamics associated with shell-model coupling schemes and in identifying the physical situations for which a given scheme is most appropriate. In particular, they suggest model Hamiltonians that are diagonal in the various coupling schemes. The dual pairing of group representations has been applied profitably in mathematics to the study of invariant theory. We show that parallel applications to the theory of symmetry and dynamical groups in physics are equally valuable. In particular, the pairing of the representations of a discrete group with those of a continuous Lie group or those of a compact Lie with those of a non-compact Lie group makes it possible to infer many properties of difficult groups from those of simpler groups. This review starts with the representations of the symmetric and unitary groups, which are used extensively in the many-particle quantum mechanics of bosonic and fermionic systems. It gives a summary of the many solutions and computational techniques for solving problems that arise in applications of symmetry methods in physics and which result from the famous Schur-Weyl duality theorem for the pairing of these representations. It continues to examine many chains of symmetry groups and dual chains of dynamical groups associated with the several coupling schemes in atomic and nuclear shell models and the valuable insights and applications that result from this examination.

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JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics.Abstract. We consider the tensor product of two irreducible unitary representations of SL2(R); in particular we obtain its reduction as a direct integral of irreducible representations. This question has been solved in certain cases by L. Pukanszky and R. P. Martin; we restate their results and also do the remaining cases. We also use the results on tensor products to strengthen slightly a result of Kunze and Stein, using the integrability properties of the coefficient functions of a representation to characterize which irreducible representations may occur in it. Introduction. The question of decomposing the tensor product of two irreducible representations of G = SL2(R) has already been dealt with in certain cases: if the two representations both belong to the complementary series, or if both belong to the principal series with even parity, or if one of them belongs to each of these classes, then the decomposition has been obtained by Pukanszky [7]. This result was extended to the case of two representations in the principal series, with any combination of parities, by Martin [6].In Section 4 we find the decomposition of the tensor product of two principal series representations, using Mackey's subgroup theorem and Frobenius reciprocity; the approach is that of Williams [10]. The mappings which display the decomposition also exist for tensor products of non-unitary principal series representations, but they need no longer be unitary isomorphisms. Fortunately, however, they are still closed maps, so with some careful analysis (Sections 5 and 6) we may apply Schur's lemma to discuss tensor products involving complementary and discrete series representations (which have realizations in the non-unitary principal series).In Section 7 we discuss tensor products of pairs of discrete series representations, using an altogether unrelated method based on the fact that they are so-called "holomorphic discrete series" representations. This approach is based on a suggestion of Roger Howe's, for which the author wishes to thank him.

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