Young tableaux

From a given Young diagram we can, in n different ways, form a "Young tableau t by filling the boxes, in any order, with the site numbers 1,2,.., n. Fig. 2 shows examples of tableaux formed from the diagrams of Fig. 1. 

The representations of U(2) are, in general, labeled by two quantum numbers. However, since we are considering only boson realizations, these representations must be totally symmetric and can thus be characterized by only one quantum number, that is, the first entry in the Young tableau... 

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... 

For example, for the four-electron case mentioned above, we take the first Young tableau... 

It is easy to figure out the relation of this with the spin of the many-electron state. It is clear that the above Young tableau corresponds to a many-electron state with the spin projection equal to... 

Transfer functions have been derived using algebraic methods based on the Hausdorff formula (Chandrakumar et al., 1986 Visalakshi and Chandrakumar, 1987), analysis in the zero-quantum frame (Muller and Ernst, 1979 Chingas et al., 1981 Chandrakumar et al., 1986), with the help of a Young tableau formulation (Listerud and Drobny, 1989 Listerud et al., 1993), and by application of Ldwdin projectors to evaluate density matrix evolutions (Chandrakumar, 1990). 

The symmetry type of F is denoted by a Young Tableau, for example... 

A Young diagram with numbers in it like (O Eq. 2.63) is called a Young tableau and it is seen at once that there is only one tableau for the last Yoimg diagram in O Eq. 2.62, namely ... 

In evaluating the elements of H and M, the special form of the spatial function may now be recognized. By assuming that electrons 1 and 2 occupy the first orbital, 3 and 4 the second, and so on, we impose a symmetry on the spatial function 0. If 0 is symmetric under transposition (12), it will be necessary to ensure that the spin factor is anrisymmetric under (12) this must be so for each doubly occupied orbital, and the first g columns of any Young tableau describing an associated spin eigenfunction will thus read... 

This problem is solved in terms of the Weyl tableaux (Fig. 7.1c) derived from the first Young tableau by inserting orbitals (or their indices) in the boxes in place of the electron indices thus, with 5 different orbitals, the products... 

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