Induced representations

In this subsection, the theory of the induced representation will be formulated in a manner similar to the treatment of Ruch and Schon-hofer 13>. For a more standard mathematical treatment, see Boemer 9>. 

If the subspaces ra are all independent, the induced representation has dimension F —fq, and we speak of the induction as regular 13>. In the mathematical literature, non-regular induction is normally not discussed, and the word induction is used for what we call regular induction 9>. We will use the notation... 

In the case of non-regular induction, one would have to express each /> in terms of a complete set for the induced representation space and then apply (29) to get the matrix elements. For general induction,... 

The right-hand side of (33) is just the dimension of the regularly induced representation [y(r> ( )], while the left-hand side is the dimension of the induced representation if the equality is satisfied in (29). This establishes our result, which may be written as... 

The induction from to Qn is the same, except that the only induced diagrams allowed are those with an even or odd number of boxes in the negative part according as the inducing representation is g or u. In Fig. 4, we give an example of the induction from Qj to S7. 

For induction from <3 7 to (3 7, we must distinguish between the indices g or u of the inducing representation and consequently ignore induced diagram pairs with an odd (for g) or even (for w) number of boxes in the minus component. Thus, referring to Fig. 4, if the inducing representation of S7 is (3,2 2) , we obtain only the representations under column (w) in the figure if it is (3,22)ff we get only those in column (g). 

For a systematic group-theoretical study of chirality functions the chemist requires a knowledge of the representation theory of symmetric and hyperoctahedral groups, and of the concept of induced representations and their properties. While excellent expositions of these topics are to be found in the mathematical literature, they are usually formulated in an idiom foreign to the chemist and are thus relatively inaccessible to him. As the present article is an attempt to bridge this mathematical gap, the theory is presented as far as possible in a unified form so as to include the cases of both achiral and chiral ligands. 

Altrnann, S.L. "Induced Representations in Crystals and Molecules" Academic Press N.Y., 1977. 

For example, consider the translation action (R, R, cr) defined above. Let V denote the complex vector space of complex-valued functions of one real variable. The induced representation of the additive group R on V is given by... 

Table 4.13. Subelements hsi gf) and MRs T(gy) of two representations of SO), TtG, obtained by the method of induced representations.

Example 4.8-3 Construct the induced representations of S(3) from those of its subgroup C(3). 

The final step going from the small IRs of the little group G(k) to the IRs of G requires the theory of induced representations (Section 4.8). At a particular k in the representation domain, the left coset expansion of G on the little group G(k) is... 

Find the representations of the space group 227 (Fd3m or O],) at the surface point B(1/2 + /3, ( I 3, Vi + a), point group Cs= E ay. [Hints-. Use the method of induced representations. Look for an isomorphism of Cs with a cyclic point group of low order. The multiplication table of Cs will be helpful.]... 

Altmann, S.L. (1977) Induced Representations in Crystals and Molecules. London Academic Press. 

Kovalev, O. V. (1993) Representations of the Crystallographic Space Groups Irreducible Representations, Induced Representations and Corepresentations, 2nd edn. Philadelphia, PA Gordon and Breach. 

The stabilizer of a vertex in a simplex, i.e. the group of all elements of S which leave a given vertex invariant, is the maximal subgroup 5 i. The set of all vertices thus will transform as the induced representation of a totally symmetric irrep of the site group in the parent group. Since this representation is certainly doubly... 

The central atom is invariant in Oh and thus transforms as Aig. The total induced representation of the function space thus is given by... 

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 ... 

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. 

As an example, in a cube one has v = 8,e = 12, f = 6, and hence 8 — 12-1-6 = 2. The 2 in the right-hand side of Eq. (6.128) is called the Euler invariant. It is a topological characteristic. Topology draws attention to properties of surfaces, which are not affected when surfaces are stretched or deformed, as one can do with objects made of rubber or clay. Topology is thus not concerned with regular shapes, and in this sense seems to be completely outside our subject of symmetry yet, as we intend to show in this section, there is in fact a deep connection, which also carries over to molecular properties. The surface to which the 2 in the theorem refers is the surface of a sphere. A convex polyhedron is indeed a polyhedron which can be embedded or mapped on the surface of a sphere. Group theory, and in particular the induction of representations, provides the tools to understand this invariant. To this end, each of the terms in the Euler equation is replaced by an induced representation, which is based on the particular nature of the corresponding structural element. In Fig. 6.8 we illustrate the results for the case of the tetrahedron. 

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