New results for the missing quantum numbers labelling the quadrupole and octupole boson basis

The many 2k-pole boson states, |NkvkαkIkMk) with k = 2, 3, realize the irreducible representation (IR) for the group reduction chains SU(2k + 1) R2k+1 R3 R2. They have been analytically studied and widely used for the description of nuclear systems. However, no analytical expression for the degeneracy d(k)v(I) of the R2k+1 IR, determined by the reduction R2k+1 R3, with k = 2, 3 is available. Thus, the number of distinct values taken by αk has been so far obtained by solving some complex equations. Here we derive analytical expressions for the degeneracy d(k)v(I) (k = 2, 3), characterizing the octupole and quadrupole boson states, respectively. The merit of this work consists of the fact that it completes the analytical expressions for the 2k-pole boson basis for k = 2, 3. The general case of R2l+1 IR representation degeneracy is also presented and a compact analytical expression for d(l)v(I) is derived