Physical Chemistry Third Edition

I Matrix Representations of Groups 1295

We can show that these matrices obey the same multiplication table as the symmetry

operations. For example,

R(̂σyz)R(̂C 2 )[−1][−1][ 1 ]R(̂σxz) (I-11)

The other two one-dimensional representations also have the same multiplication table.

If a matrix has all zero elements except for those in square areas along the principal

diagonal the matrix is said to be ablock-diagonal matrix. An example of a block-

diagonal 5 by 5 matrix is

F

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

10000

02300

04300

00056

00078

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

(I-12)

This matrix containsa1by1block and two 2 by 2 blocks. If two matrices of the same

size that are block-diagonal with blocks of the same sizes in the same positions are

multiplied together, the product matrix is block-diagonal with blocks of the same sizes

in the same positions.

If a representation of some group consists of a set of matrices that are all block-

diagonal with blocks of the same sizes in the same positions, it is a reducible repre-

sentation. A new set of matrices obtained by taking the corresponding block out of

each matrix is also a representation of the group. Even if a representation consists of

matrices that are not block-diagonal, it is a reducible representation if a similarity trans-

formation produces matrices that are all block-diagonal in the same way. Asimilarity

transformationon the matrixBmeans the carrying out of two matrix multiplications

with some matrix and its inverse as follows to yield a new matrixC:

CA−^1 BA (I-13)

The matrixAcan be any square matrix of the same size asB. If the same similarity

transformation is carried out on every matrix in a representation, the new set of matrices

is also a representation. If the same similarity transformation when carried out on every

matrix in a representation produces a set of matrices that are all block-diagonal with the

same size blocks in the same order, then the original representation is reducible, and

each set of corresponding blocks in the new set of matrices forms a new representation.

If no such transformation can be found, then the original representation is irreducible.

I.2 Classes in a Group

A group can often be divided into classes. IfAandBare both members of the same

group the matrixCthat results from the similarity transformation of Eq. (I-13) must be

a member of the group. If the membersAandA−^1 are replaced in turn by every other

member of the group and its inverse, keepingBfixed, it is found in many cases that only

part of the members of the group will occur in the place ofC. These members constitute

aclasswithin the group and if any one of them is put in place ofBin Eq. (I-13) only

members of the class will result in the place ofC. Every operator of theC 2 vgroup is

in a class by itself.