A direct product representation of a group is a representation with each element being the Kronecker product of the elements of two other irreducible representations and of corresponding to the same symmetry operation .

For example, if the dimensions of and are 2 and 3 respectively, i.e. and , then

In general, if the dimensions of and are and respectively, then the dimension of is :

###### Question

Show that

###### Answer

With reference to eq75, let’s denote the block by . So,

If is also a representation of , then it must be consistent with the closure property of , i.e. . The proof is as follows:

where the 2^{nd} equality uses the identity proven in the Q&A above.

With reference to eq75, the trace of is

It follows that the direct product of three representations is

which can be extended to direct products of more than three representations.

###### Question

With reference to the point group, show that , while can be decomposed into the direct sum of .

###### Answer

Using eq76, we include and in the character table of the point group as follows:

Clearly, , which implies that is an irreducible representation of . Since the number of irreducible representations of a point group is equal to the number of classes of the group, is a reducible representation because its characters are not equivalent to the characters of any of the 3 irreducible representations. We then use eq27 for the decomposition of , where

Therefore, .

Finally, consider the sets of linearly independent functions

i) , where

ii) , where

iii) , where

that transform according to , and respectively.

Since , and are sets of linearly independent functions, we can write

According to a previous article, we can also write

where is a symmetry operation of and , and are the matrix entries of , and respectively.

###### Question

What is the relation between the matrix entries of , and in ?

###### Answer

Let the matrix entries of , and be , and respectively, where

Using the ordering convention called dictionary order, where the order of or is given by

or in terms of the notation or ,

In other words, is determined by and , and is determined by and . For example, if and ,

We can then express the matrix entries of as

Multiplying eq77a by eq77b,

From eq77g, we have and we can rewrite the RHS of eq77h as , where . Hence, eq77h is equivalent to eq77c. Combining eq77f and eq77h, we have

This implies that if the functions and are bases for the irreducible representations of and of a point group respectively, then is the basis for , which is the direct product of and . In other words,

*If the functions ** and ** transform according to the irreducible representations of ** and * * of a point group respectively, then the function **, which is the product of the functions ** and ** must transform according to the direct product of **.*

###### Question

Show that the reducible representation of the direct product of two irreducible representations and contains the totally symmetric representation if and only if is the complex-conjugate representation of .

###### Answer

Using eq27a, the number of times the totally symmetric representation appears in the decomposition of is

Substitute eq76 in eq78

Comparing eq79 with eq22, if and only if and vanishes otherwise.