Group theory representation reduction

Thus, any representation Tcan be expressed as a function of its irreducible representations Pi. This operation is written as P = S a, Pi, where a, indicates the number of times that Pi appears in the reduction. In group theory, it is said that the reducible representation P is reduced into its Pi irreducible representations. The reduction operation is the key point for applying group theory in spectroscopy. To perform a reduction, we need to use the so-called character tables. 

To tackle this problem, we first need to know how a given representation F is reduced to its irreducible representations T) in other words, to determine the coefficients a, in the equation F= Sat Fi. Although this is a key problem in group theory, here we only explain how to perform this reduction without entering into formal details, which can easily be found in specialized textbooks. 

The previous example has shown how group theory can be used in a symmetry reduction problem. This symmetry reduction also occurs when an ion is incorporated in a crystal. We will now treat how to predict the number of energy levels of the ion in the crystal (the active center) and how to properly label these levels by irreducible representations. 

N. Chriss and V. Ginzburg, Representation theory and complex geometry (Geometric techniques in representation theory of reductive groups) , Progress in Math. Birkhauser, (to appear). 

In applying the methods of group theory to problems related to molecular structure or dynamics, the procedure that is followed usually involves deriving a reducible representation for the phenomenon of interest, such as molecular vibration, and then decomposing it into its irreducible components. (A reducible representation will always be a sum of irreducible ones.) Although the decomposition can sometimes be accomplished by inspection, for the more general case, the following reduction... 

It is important to note that no symmetry operation exchanges an axial ligands with an equatorial one. As these two types of ligands are therefore non-equivalent, both chemically and according to group theory, they can he considered separately. The characters of the representations Per (eq) and P (ax) are given in Table 6.22. From the reduction formula (6.5), we find ... 

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