Energy expressions for n=3 and 4 systems in a single-j shell

For systems with three and four fermions within a single-j shell, analytical expressions for the state energies are presented from a decomposition of the angular momentum. In some important cases the expressions acquire a very simple form. The expression may help us in understanding the structure of isomeric states. The decomposition also makes it possible to construct the algebraic condition for conservation of seniority.