The regular representation of a group is a reducible representation that is generated from a rearranged multiplication table of the group. It is used to derive an important property (see eq40 below) for constructing character tables.

Consider the re-arranged multiplication table for the point group such that all the identity elements are along the diagonal:

An element of the regular representation of the group is derived from table II in the form of a matrix, whose entries are 1 when the element of occurs in table II and zero otherwise. For example,

In other words, we have

where is the -th row and -th column matrix entry of the -th element of the representation of .

###### Question

Show that each matrix of the regular representation of the point group has an inverse.

###### Answer

The entry 1 appears only once in every row (or column) of a matrix, e.g. , of the regular representation. We can therefore swap the rows (or columns) of to form . According to determinant property 1, and consequently, according to determinant property 8, . Therefore, is non-singular, according to determinant property 11. The same logic applies to the rest of the matrices.

If these derived matrices are truly elements of a representation of , then they must satisfy the closure property of , i.e.

where

Therefore, in eq35 needs to be equal to to satisfy eq32. To prove this, we multiply the condition in eq31 on the left by to give . Similarly, we multiply the condition in eq34 on the left by and then on the right by to give . Combining the two results, we have , which when multiplied on the right by and then on the left by , gives , which completes the proof.

By inspecting table II, the character of the regular representation is

###### Question

Show that the regular representation is reducible.

###### Answer

If the regular representation is reducible, the LHS of eq28 must be greater than . Applying the LHS of eq28 to the regular representation and using eq36,

For non-single element groups, and hence .

Finally, we shall prove that . From eq25 and eq27, we have

where and are the characters for the -th class of the regular representation and irreducible representation respectively.

Expanding eq38 and using eq36,

###### Question

Why is ?

###### Answer

The regular representation for is a reducible representation that is the direct sum of irreducible representations under the class . Each of these constituent irreducible representations is in general a matrix with trace of .

Eq39 therefore states that each constituent irreducible representation occurs in the regular representation a number of times that is equal to the dimension of the corresponding irreducible representation. For example, with reference to the table below, , where , and . In other words, each of the irreducible representations and appears once in the regular representation matrix element , while the irreducible representation appears twice.

Since the matrix dimension for each element of the regular representation is ,

where we have used eq39 and where refers to the dimension of the -th irreducible representation of a group.