Irreducible representations of $SU(3)$ for nucleons in a oscillator shell $η$ with $η\le 8$: Application to nuclei with proxy-$SU(4)$ symmetry

Applying some simple rules, established in group theory literature, generated are codes (easy to modify as required) for obtaining $SU(3)$ irreducible representations (irreps) for a given number of nucleons in a given oscillator shell $η$ with $η\le 8$. In this approach, first the $SU(3)$ irreps are obtained for spinless fermions in a given $η$ shell. Using the codes developed, tables of $SU(3)$ irreps are generated for identical nucleons (protons or neutrons) with a given total spin $S$. Similarly, the $SU(3)$ irreps are generated for nucleons with a given spin-isospin $SU(4)$ irrep. Some applications of these tables for $η=4$ shell for $52 \le \mbox{Z,N} \le 78$ nuclei with proxy-$SU(4)$ symmetry are presented. Also, the procedure used for the shell model $SU(3)$ limits is easy to be extended for the $SU(3)$ limit of various interacting boson models and some examples for these are also presented.