The decomposition of a reducible representation of a group reduces it to the direct sum of irreducible representations of the group. We have shown in an earlier article that this involves decomposing a block diagonal matrix into the direct sum of its constituent square matrices. In this article, we shall derive some useful equations involving the characters of the representations of a group (see eq25, eq27a, eq29 and eq30 below).

Consider an element of a reducible representation that has undergone a similarity transformation to the following block diagonal matrix:

where , , and .

Clearly, . Since the trace of a matrix belonging to the -th class of the -th irreducible representation is called the character of a class , we have . However, a particular constituent irreducible representation may appear more than once in the decomposition. Therefore,

where is the number of times appear in the direct sum.

Multiplying eq25 by , where is the number of elements in the -th class, and sum over ,

Substituting eq23 in eq26,

Since ,

Eq27 is sometimes written as a sum over the individual symmetry operations rather than over the various classes:

###### Question

Show that .

###### Answer

The complex conjugate of eq25, with a change of the dummy index from to , is . Multiply this by eq25 and by and sum over ,

Swap the dummy indices of eq23 to , which when substituted in the above equation, gives

Since ,

With reference to eq25 and eq28, if one of the is equal to 1, with the rest of the equal to zero, then is an irreducible representation. This implies that

###### Question

Show that if two reducible representations and of a group are equivalent, they decompose into the same direct sum of irreducible representations of .

###### Answer

Since and are related by a similarity transformation, for every . Using eq27a,