Spin projection of determinants

We assume that the determinant is written as a product of alpha creation operators, an alpha string, times a product of beta creation operators, a beta string, working on the vacuum state  

The operators corresponding to the doubly occupied orbitals can be moved out in front of the other creation operators. The determinant then appears as a product of three strings - a core string, an alpha string and a beta string  

To obtain the eigenvalue of the projected-spin operator, we first calculate the commutators between and the strings (2.4.15)-(2.4.17). Using (2.3.13), these commutators become 

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