Total spin of determinants

We now proceed to determine the effect of the operator of the total spin on the determinants  

For this purpose, we shall need the commutators between the operator strings (2.4.15)-(2.4.17) and the shift c )eratcM. The commutators involving the raising operator are given by 

The effect of the lowering operator on the alpha string is to generate a linear combination of n strings, each obtained by lowering the spin projection of one cM bital in the alpha string. 

To evaluate (2.4.26), we first apply the raising operator to the determinants. The raising operator generates a linear combination of determinants, as seen from the expression 

Invoking the Jacobi identity (1.8.17), the double commutator may be written as 

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