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This article reviews many manifestations and applications of dualrepresentations of pairs of groups, primarily in atomic and nuclear physics.Examples are given to show how such paired representations are powerful aids inunderstanding the dynamics associated with shell-model coupling schemes and inidentifying the physical situations for which a given scheme is mostappropriate. In particular, they suggest model Hamiltonians that are diagonalin the various coupling schemes. The dual pairing of group representations hasbeen applied profitably in mathematics to the study of invariant theory. Weshow that parallel applications to the theory of symmetry and dynamical groupsin physics are equally valuable. In particular, the pairing of therepresentations of a discrete group with those of a continuous Lie group orthose of a compact Lie with those of a non-compact Lie group makes it possibleto infer many properties of difficult groups from those of simpler groups. Thisreview starts with the representations of the symmetric and unitary groups,which are used extensively in the many-particle quantum mechanics of bosonicand fermionic systems. It gives a summary of the many solutions andcomputational techniques for solving problems that arise in applications ofsymmetry methods in physics and which result from the famous Schur-Weyl dualitytheorem for the pairing of these representations. It continues to examine manychains of symmetry groups and dual chains of dynamical groups associated withthe several coupling schemes in atomic and nuclear shell models and thevaluable insights and applications that result from this examination.

Dual pairing of symmetry groups and dynamical groups in physics