Direct product representation

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 :



Show that


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 2nd 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.



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


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.



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


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 .



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 .


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.



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