  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.
