Subduced Representations

We now consider the representation r of induced by the irreducible representation y of . We pose the question when F is broken up into its irreducible parts, how many times will each irreducible representation rw appear To decide this, we first observe that the independent aj > which form a basis for F can all be generated from a single j > of r by application of the elements of , including those of. For the vectors of r, this follows from the irreducibility of y, for the others from the nature of the induction process. Since the e-operators form a complete set in U, one can equally well say that the basis vectors of r are generated by applying all the e-operators to a single j >. Now suppose that y appears c times in the subduced representation TW ( ) . Choose a basis for such that the first c basis vectors transform like j > under . In this basis we have... 

For Z may be zero, but need not be. It follows that we can generate from j > at most c vectors transforming like the fc th basis vector of fM. When the operations of are applied to these, each may separately generate the representation I M, but it cannot be generated more times than this. Thus, the representation fW appears in T at most the same number of times that y appears in the subduced representation of r( ), and its decomposition into irreducible components is... 

To analyze the implications of this, we choose a basis for each irreducible representation, T of S such that the first zr basis functions transform according to under , where zr is the number of times r[Pg.48]

To get the characters for the representation of subduced by a given representation of S, we just copy down the characters of that representation for the elements of which are also in . This is done for the irreducible representations of 4 subduced onto D2< in Table 3. Comparing Tables 2 and 3, and using the standard formula for finding the irreducible parts of a representation by means of the characters, we see that the representations subduced by. T<8> and /1<5) contain / ( )... 

II) If G contains a subgroup H then the choice of the set of poles made for G should be such that rule I is still valid for H, otherwise the representations of G will not subduce properly to those of H. Subduction means the omission of those elements of G that are not members of H and properly means that the matrix representatives (MRs) of the operators in a particular class have the same characters in H as they do in G. 

In a separate paper, Fujita jointly with Sherif El-Basil57 used the concept of doubly-colored graphs to visualize the abstract concepts such as subductions of coset representations, double cosets, and unit-subduced-cycle-indices,58 which had been mathematically formulated in the framework of coset algebraic theory developed by Fujita.42... 

For a proof of this theorem, we refer to the literature [19, 20]. The theorem is not only applicable to molecular vibrations but is also directly in line with the LCAO method in molecular quantum chemistry. In this method the molecular orbitals (MOs) are constructed from atomic basis sets that are defined on the constituent atoms. An atomic basis set, such as or 4/, corresponds to a fibre, emanating, as it were, from the atomic centre. Usually, such basis sets obey spherical symmetry, since they are defined for the isolated atoms. As such, they are also invariant under the molecular point group [21]. As an example, a set of 4/ polarisation functions on a chlorine ligand in a RhClg complex is itself adapted to octahedral symmetry as 2 + tiu + tiu This representation thus corresponds to V. In the C4 site symmetry these irreps subduce ai-ybi- -b2- -2e. According to the theorem, theLCAOs based on the 4/ orbitals thus will transform as ... 

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