Frobenius reciprocity theorem

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 ... 

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