Irreducible/irreducibility, generally

More generally, it is possible to combine sets of Cartesian displacement coordinates qk into so-called symmetry adapted coordinates Qrj, where the index F labels the irreducible representation and j labels the particular combination of that symmetry. These symmetry adapted coordinates can be formed by applying the point group projection operators to the individual Cartesian displacement coordinates. 

A components. Analogous projections in the E and A2 directions give components of 1 and 0, respectively. In general, to determine the number nr of times irreducible... 

The tables of characters have the general form shown in Table 5. Each colipua represents a class of symmetry operation, while the rows designate the different irreducible representations. The entries in the table are simply the characters (traces) of the corresponding matrices. Two specific properties of the character tables will now be considered. 

Before going on to consider applications of group theory in physical problems, it is necessary to discuss several general properties of irreducible representations. First, suppose that a given group is of order g and that the g operations have been collected into k different classes of mutually conjugate operations. It can be shown that the group Q possesses precisely k nonequivalent irreducible representations, T(1), r(2).r(t>, whose dimen-... 

As indicated in Section 3.4, the integral of an odd function, taken between symmetric limits, is equal to zero. More generally, the integral of a function that is not symmetric with respect to all operations of the appropriate point group will vanish. Thus, if the integrand is composed of a product of functions, each of which belongs to a particular irreducible representation, the overall symmetry is given by the direct product of these irreducible representations. 

The product representation D XI/ is in general reducible. It can be decomposed into its irreducible components... 

A similar problem was discussed in Section IV-C, where the motion of a single ion in the solvent was considered. We shall briefly indicate in Appendix A2 how this calculation may be generalized to the case where the heavy particle has a nonvanishing wave number. The resolvant operator is decomposed into irreducible diagonal fragments, which are then expanded in powers of yx the result is ... 

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