Additional classification of terms in isospin basis

In Chapter 14 we have already discussed the group-theoretical method of classification of the states of a shell of equivalent electrons. Remembering that second-quantization operators in isospin space have an additional degree of freedom, we can approach the classification of states in isospin basis in exactly the same way. 

These operators are generators of the Usi+4 group, and the characteristics of irreducible representations of the latter correspond only to the number of electrons N in the (ll)N configuration. The possible subgroup chains that preserve the classification according to the quantum numbers L, S and T are given by the following reduction schemes  

The other subgroup chain in (18.45) includes the Sp4i+2 group, whose generators are the tensors with odd sum of ranks k + k. Since 

Separating the operators that are scalars in orbital and spin spaces 

These operators play the role of generators of the five-dimensional quasispin group Sp4, which can be readily verified by comparing their commu- 

The algebra of the Sp4 group coincides with the algebra of the rotation group of five-dimensional Euclidean space R, i.e. these groups are locally isomorphic. The irreducible representations of the Sp4 group can be characterized by a set of two parameters (v, t) where the seniority quantum number v for the five-dimensional quasispin group indicates the number 

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