Algebra tensor representation

The next important problem in algebraic theory is the construction of the basis states (the representations) on which the operators X act. A particular role is played by the irreducible representations (Appendix A), which can be labeled by a set of quantum numbers. For each algebra one knows precisely how many quantum numbers there are, and a list is given in Appendix A. The quantum numbers are conveniently arranged in patterns (or tableaux), called Young tableaux. Tensor representations of Lie algebras are characterized by a set of integers... 

Table A.4 Number of integers that characterize the tensor representations of Lie algebras...

We return to the simple example of the angular momentum algebra, SO(3). Its tensor representations are characterized by one integer (Table A.4), that is, the angular momentum quantum number J. Similarly, the representations of SO(2) are characterized by one integer (Table A.4) that is, M the projection of the angular momentum on the z axis. The complete chain of algebras is... 

The presence of two nuclear spins means that there is considerable choice in the selection of basis functions the reader who wishes to practice virtuosity in irreducible tensor algebra is invited to calculate the matrix elements in the different coupled representations that are possible In fact the sensible choice, particularly when a strong magnetic field is to be applied, is the nuclear spin-decoupled basis set t], A N, S, J, Mj /N, MN /H, MH). Again note the possible source of confusion here MN is the space-fixed component of the nitrogen nuclear spin /N, not the space-fixed component of N. This nuclear spin-decoupled basis set was the one chosen by Wayne and Radford in their analysis of the NH spectrum. 

Several remarks are necessary. The units of the corresponding group algebra are provided with the superscript [w] in order to indicate that their representation is realized in the space of tensors of rank m, respectively. The first step is to use Equation [64] in order to generate just one tensor for the pGj-dimensional irreducible representation. In fact, the number of free parameters must coincide with the... 

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