Dimension of representations

Exercise 13.4-5 Show that the dimension / of representations of the third kind (c) is an even number. 

Exercise 13.4-6 It was shown in Exercise 13.4-5 that the dimension / of representations of type (c) is an even integer. Therefore, even though time reversal introduces no new degeneracies, l is always at least 2 and Kramers theorem is satisfied. 

In QSAR modeling, the question of molecular representation is central. For the modeling, a molecule is represented as a multidimensional vector, i.e., a molecule is a point in multidimensional representational space. An ideal representation should be unique, uniform, reversible, and invariant on rotation and translation of molecules. Unique means that different structures give different representations, uniform means that the dimension of representation is the same for all structures, reversible means that the structure can be unambiguously reconstructed from the representation vector. Furthermore, invariant means that the representation is not sensitive if a molecule is rotated or translated. It is not expected that we would find a general representation that fulfills all requirements simultaneously [26]. Nowadays, thousands of descriptors and structural representations are in use [26-29]. We will give a short overview, and references about descriptors that often appear in QSAR studies related to mutagenicity of aromatic amines are presented in more detail. 

In general, the result of this procedure for any irreducible representation in any group will be the order of the group divided by the dimension of the representation. That is, if h is the order of the group, f is the dimension of representation fj, and TV (R)... 

The treatment of translational motion in three dimensions involves representation of particle motions in tenns... 

Figure 6.15 (a) Structure of BaHg showing Cj symmetry (b) dimensions and representation of the bonding... 

Case (b) aproduces a set of functions t/rj, which is independent of the set but which also forms a basis for A (u) of H. The irreducible co-representation D of G corresponds again to a single irreducible representation of H, but has twice its dimension. In this case the dimension of A (u) is doubled. 

Fig. 75.—Vectorial representation in two dimensions of a freely jointed chain. A random walk of fifty steps.

As shown earlier, the operation is always in a class by itself, as it commutes with all other operations of the group. It is identified with Ti, the arbitrarily chosen first class of operation In a given representati.Qnt ie operation E corresponds to a. unit matrix whose order is equal to the dimension of the representation. Hence, the esultipg character, the sums of e diagonal elements, is also equal to the dimension of the representation, The dimension of each representation can thus be easily determined by inspection of the corresponding entry in the first column of characters in the table. 

This statement is often taken as a basic theorem of representation theory. It is found that for any symmetry group there is only one set of k integers (zero or positive), the sum of whose squares is equal to g, the order of the group. Hence, from Eq. (29), the number of times that each irreducible representation appears in the reduced representation, as well as its dimension, can be determined for any group. 

This is an immediate consequence of the lowering of the symmetry as, even in the regular octahedral geometry, group theory tells us that the highest dimension of the irreducible representation is three. This is the basis of Crystal Field Theory, whose deeply symmetry-based formalism was developed by Bethe in 1929 [16]. 

It applies for both formulations above that the expansion in principle contains an infinite number of terms. The convergence to a few lowest order terms relies on the ability to orderly separate influences of the dominant rf irradiation terms (through a suitable interaction frame) from the less dominant internal terms of the Hamiltonian. In principle, this may be overcome using the spectral theorem (or the Caley-Hamilton theorem [57]) providing a closed (i.e., exact) solution to the Baker-Campbell-Hausdorf problem with all dependencies included in n terms where n designates the dimension of the Hilbert-space matrix representation (e.g., 2 for a single spin-1/2, 4 for a two-spin-1/2 system) [58, 59]. 

The dimension of a representation is the same as the order of the matrix. To reduce a representation it is necessary to reduce its order. It is noted that the dimension of a matrix representation corresponds to the character of the identity (E) matrix. 

A vector can be thought of as a point in -dimensional space, although the graphical representation of such a point, when the dimension of the vector is greater than 3, is not feasible. The general rules for matrix addition, subtraction, and multiplication described in Section A.2 apply also to vectors. 

Due to table-storage limitations, the applicability of pre-computed lookup tables will be limited by the dimensions of the allowable region. Standard pre-computed lookup tables (i.e., ones that do not attempt to find a low-dimensional representation of the chemical kinetics) will be limited by computer memory to three to five chemical species. For example, five scalars on a reasonably refined grid yields ... 

Familiarity is also assumed with the concepts of representation and irreducible representation (IR). A representation r of dimension n associates to each group element s an n X n matrix D(s), with matrix elements D(s)y, in such a way that for every s, t, D(s)D(t) =D(st), with the product formed by ordinary matrix multiplication. We will sometimes use the bra-ket notation... 

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... 

A standard tableau is defined as one in which the numbers increase when one reads from left to right in each row and from top to bottom in each column. It can be shown Mi) that the dimension of a representation is equal to the number of standard tableaux associated with the corresponding diagram. The reader is also referred to the literature Mi) for methods of calculating characters of the representations from the diagrams. 

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