Lie Group of the Strong-Coupling Theory. III. Dynamical Generation of Extra Internal Symmetry for a Class of Representations

The strong-coupling dynamics, with isospin $\mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_I$ as the given internal symmetry group, leads to the generation of a larger internal symmetry group $\mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_{I_1}\otimes \mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_{I_2}$ (${\mathbf{I}}_1$+${\mathbf{I}}_2$=I) for the hyperon isobar-pion coupling constants, the hyperon isobar series having isospin-spin $I=J\pm \frac{1}{2}=0, 1, 2, \cdots , \infty$. In this connection we describe a "modified contraction" method for constructing the irreducible representations of the strong-coupling group $\mathcal{G}=[SU{\mathrm{}\left(2\right)\mathrm{}}_I\otimes \mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_J]\times T_9$, which starts from the "uncontracted" group $\mathrm{SU}\left(2\right)\mathrm{}\otimes \mathrm{SU}\left(4\right)\mathrm{}$ and provides a transparent formulation of the dynamical generation of extra symmetry for a class of irreducible representations of group $\mathcal{G}$. One more method for constructing the representations of the group $\mathcal{G}$ is also described which starts from a knowledge of the irreducible representations of inhomogeneous Euclidean group in four dimensions.