Reducible representations The orthogonality theorem

In this table the notation (x, y) indicates that the pair (x, y) transforms into linear combinations of x and y under the operations of the group. As usual, is the totally symmetric representation. 

All of these direct products of the irreducible representations are themselves irreducible representations. On the other hand when we form the direct product e x e,WQ obtain for the characters  

These numbers do not correspond to the characters of any of the irreducible representations. It follows that the direct product e x e belongs to a reducible representation. The character of any reducible representation is the sum of the characters of the irreducible representations that compose it. In this particularly simple case we can see by inspection of the character table that 

We say that the direct product e x e contains the irreducible representations ai,d2 and e. 

In more complicated cases, inspection is not a practical way to proceed. For these we use the orthogonality theorem, which we must simply state without proof. This theorem can be written in the special form 

---