Induction Revisited The Fibre Bundle

In Chap. 4 we left induction after the proof of the Frobenius reciprocity theorem. In that proof the important concept of the positional representation was introduced. This described the permutation of the sites under the action of the group elements. Further, we defined local functions on the sites which transformed as irreps of the site symmetry. As an example, if we want to describe the displacement of a cluster atom in a polyhedron, two local functions are required a totally-symmetric one for the radial displacement and a twofold-degenerate one for the tangential displacements. In cylindrical symmetry, these are labelled a and tt, respectively. The mechanical representation, i.e. the representation of the cluster displacements, is then the sum of the two induced representations  

This is precisely the set of fluorine displacements that we constructed in Sect. 4.8 in order to describe the vibrational modes of UFe. One remarkable result of induction theory is that the mechanical representation can also be obtained as the direct product of the positional representation and the translational representation, T u, this is the representation of the three displacements of the centre of the cluster. 

It is as if the displacements of the central point of the octahedron were relocated to every ligand site. The elementary function space of the displacements of the central atom, which transforms as the translational irrep, Tu, is called the standard fibre. This fibre is attached to every site of the cluster, and the set of these fibres is the fibre bundle. The action of the group permutes fibres of the bundle. The following induction theorem holds  

Theorem 14 Consider a standard fibre, consisting of a function space that is invariant under the action of the group. In a cluster of equivalent sites, we can form a fibre bundle by associating this standard fibre with every site position. The induced representation of the fibre bundle is then the direct product of the irrep of the standard fibre with the positional representation. 

For V being the representation of the standard fibre, 7i in our example, and V the positional representation of the set of equivalent sites in the molecule, one has 

---