Young tableau operator

For fermions, the projection operators are simply Young operators, derived from the appropriate Young tableau, as will be shown below. 

Projection operators for irreducible representations of the symmetric group are obtained easily from their corresponding Young tableaux. A Young tableau is created from a Young frame. A Young frame is a series of connected boxes such as... 

Combining the operators P, Q, we obtain the Young operator" associated with the tableau t ... 

The Young operator Y in (1) antisymmetrizes with respect to permutations of sites in the same column in its tableau. The monomial yi, therefore, cannot be symmetrical with respect to any two sites in the same column, i.e., it cannot contain the same power of A for any two such sites. The powers of A for the sites in a given column, therefore, must all be different. The lowest possible choice consistent with this is that they be 0, 1, 2,. .., p, for a column of length /t. Thus, y can be chosen to be independent of A for sites in the first row of the tableau, and to contain A for sites in the second row, A2 for those in the third, etc. The total order is therefore... 

Young defined two operators, the row symmetrizer V and the column antisymmetrizer A/ , and we assume the these operate on (permute) the particle labels not the orbital labels. Each tableau designates a product of orbitals with a particular ordering... 

First we define the Young operators. For any tableau it is possible to define a Young operator Ej. that, like the antisymmetrizer A in (3.1.7), is a linear combination of permutations (the group elements) and is thus an element of the group algebra . In dealing with a particular tableau, the labels [A] and r may be suppressed and E may be expressed as... 

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