Character of a representation

Since the characters of a representation must be uniquely defined, we see that those we would obtain by the above procedure when J is half-integral cannot belong to true representations. 

In particular, the character of a representation is the same as its block form (with the maximum number of blocks that correspond to the irreducible representations) ... 

Information on the irreducible representations of the various point groups is presented customarily in tables that are called character tables because they give the character of the irreducible representatimi of each symmetry operation in a point group. The character of a representation is the trace (the sum of the diagonal elements) of the matrix that represents that operatimi. Character tables for the D2h, C2v and 4 point groups are presented in Tables 4.2, 4.3 and 4.4. The symmetry elements of the point group are displayed in the top row of each table, and the conventional names, called Mulliken symbols, of the irreducible representations are given in the first column. Letters A and B are used as the Mulliken symbols for... 

Whenever a fiinction can be written as a product of two or more fiinctions, each of which belongs to one of the synnnetry classes, the symmetry of the product fiinction is the direct product of the syimnetries of its constituents. This direct product is obtained in non-degenerate cases by taking the product of the characters for each symmetry operation. For example, the fiinction xy will have a symmetry given by the direct product of the syimnetries of v and ofy this direct product is obtained by taking the product of the characters for each synnnetry operation. In this example it may be seen that, for each operation, the product of the characters for Bj and B2 irreducible representations gives the character of the representation, so xy transfonns as A2. 

Consider the trans isomer of butadiene. Both the symmetry operations that define the group < 2h and the characters of the representation r are given in Table 3. The reduction of this representation leads to Tn =2Bg 2Aa. Thus, two linear combinations of the atomic orbitals can be constructed of symmetry Bg and two others of symmetry A. Their use will factor the secular determinant into two 2x2 blocks, as described in the following paragraph. 

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 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. 

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 / ( )... 

In Table 7.5, we show the character (defined as the set of character elements of a representation) of different representations (from / = 1 to 6) of the 0 group. The character elements were obtained from Equation (7.7). These representations, which were irreducible in the full rotation group, are in general reducible in 0, as can be seen by inspecting the character table of 0 (in Table 7.4). Thus, the next step is to decompose them into irreducible representations of 0, as we did in Example 7.1. Table 7.5 also includes this reduction in other words, the irreducible representations of group O into which each representation is decomposed. We will use this table when treating relevant examples in Section 7.6. 

If we consider the sets of functions (xi, ti, zi) and (xa, yi, zf), both of which belong to the same representation T of the <9 group, the product functions constitute a nine-dimensional space. Consequently, these product functions belong to a nine-dimensional representation, denoted by Ti x Ti. It can be shown that the character of this representation is given by =... 

Equation (36) is derived by requiring that the sum of the characters of the representations of Ok which correspond to for a fixed class of Ok be equal to the character of the corresponding class of i2(3). In this way sufficient equations are obtained to determine the number of times each representation of Ok occurs in Equation (35). Each rotation by an angle 0 about an axis in space forms a class of E(3). The character of an L dimensional, irreducible representation of R 3) is... 

An extremely useful mode of calculation with characters arises from considering the characters of any representation as a vector. Thus the four character vectors of C2v are ... 

In the text, when the character of a set of orbitals is deduced to give a reducible representation, the reduction to a sum of irreducible representations has been carried out by inspection of the appropriate character table. In some instances this procedure can be lengthy and unreliable. The formal method can also be lengthy, but it is highly reliable, although not to be recommended for simple cases where inspection of the character table is usually sufficient. The formal method will be explained by doing an example. 

In this section we define characters. Associated to each finite-dimensional representation (G, V, p) is a complex-valued function on the group G, called the character of the representation Recall the trace of an operator (Definition 2,8) the sum of the diagonal elements of the corresponding matrix, expressed in any basis. 

This character is the same as the character of the representation on by matrix multiplication in fact, these two representations are isomorphic, as the reader may show in Exercise 4.36. This is an example of the general phenomenon that will help us to classify representations finite-dimensional representations are isomorphic if and only if their characters are equal. See Proposition 6.12. Note that while a representation is a relatively complicated object, a character is simply a function from a group to the complex numbers it is remarkable that so much information about the complicated object is encapsulated in the simpler object. 

The character of a dual representation is the complex conjugate of the character of the original. 

Proof. We calculate the scalar product by constructing a linear operator P whose trace is equal to the scalar product. Consider the representation (G, HomCVi, V2), cr) dehned in Proposition 5.12. Let / denote the character of this representation. By Proposition 5.14 we know that / = XiX2- Consider the linear operator... 

Consider the character of the representation / . Recall from Proposition 4.8 that there is a polynomial such that... 

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