Group algebraic representation of the antisymmetrizer

As we have seen in Eq. (5.21), an element of the group may be written as a sum over the algebra basis. For the symmetric groups, this takes the form, 

We wish to apply permutations, and the antisymmetrizer to products of spin-orbitals that provide a basis for a variational calculation. If each of these represents a pure spin state, the function may be factored into a spatial and a spin part. Therefore, the whole product, 4, may be written as a product of a separate spatial function and a spin function. Each of these is, of course, a product of spatial or spin functions of the individual particles, 

In line with the last section we give a version of Eq. (5.89) using the non-orthogonal matrix basis. 

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