Conservation laws in the $1f_{7/2}$ shell model of ${}^{48}\mathrm{Cr}$

Conservation laws in the $1f_{7/2}$ shell model of ${}^{48}\mathrm{Cr}$ found in numeric studies by Escuderos, Zamick, and Bayman [arXiv:nucl-th/0506050 (2005)] and me [K. Neergård, Phys. Rev. C 90, 014318 (2014)] are explained by symmetry under particle-hole conjugation and the structure of the irreps of the symplectic group Sp(4). A generalization is discussed.

Figure 1

Weighs of the $\text{Sp}\left(4\right)$ irrep $\left[\lambda \mu \right]=\left[83\right]$. Horizontal or vertical neighbors have distances one. The rectangle is the border of the range (11).

Figure 2

Weight diagrams of the $\text{Sp}\left(4\right)$ irreps corresponding for $j=7/2$ to the $\text{Sp}\left(2\Omega \right)=\text{Sp}\left(8\right)$ irreps $vt=00$, 40, 80, and 61. Rings indicate weights $({\mathcal{S}}_n,{\mathcal{S}}_p)$ with integral coordinates.