Simple unified proofs of four duality theorems

Duality relationships between the irreps (irreducible representations) of pairs of distinct commuting groups, \documentclass[12pt]{minimal}\begin{document}$G_1$\end{document} G 1 and \documentclass[12pt]{minimal}\begin{document}$G_2$\end{document} G 2 , on Hilbert spaces of interest have long played important roles in the atomic and nuclear shell models. In addition to the well-known Schur–Weyl duality, the most widely used duality relationships are the so-called: unitary–unitary, orthogonal–symplectic (i.e., noncompact symplectic), symplectic–symplectic (compact symplectics), and orthogonal–orthogonal dualities. Proofs of these dualities exist in the literature. But most of them are not readily accessible to physicists or give little insight into how they might be used in practice. This paper presents unified proofs of the above-mentioned dualities based on the explicit construction of states which are simultaneously of extreme weight for the actions of both \documentclass[12pt]{minimal}\begin{document}$G_1$\end{document} G 1 and \documentclass[12pt]{minimal}\begin{document}$G_2$\end{document} G 2 . The proofs expressed in language familiar to physicists are simple, systematic, and provide useful insights.