is uniquely determined. It is called the canonical decomposition. Let be the set of all irreducible representations of a group up to isomorphism. Let be a representation of and let be the set of all isotypes of The projection corresponding to the canonical decomposition is given by where and is the character belonging to In the following, we show how to determine the isotype to the trivial representation: Definition (Projection formula). For every representation of a group we define In general, is not -linear. We define Then is a -linear map, because Proposition. The map is a projection from to This proposition enables us to determine the isotype to the trivial subrepresentation of a given representation explicitly. How often the trivial representation occurs in is given by This result is a consequence of the fact that the eigenvalues of a projection are only or and that the eigenspace corresponding to the eigenvalue is the image of the projection. Since the trace of the projection is the sum of all eigenvalues, we obtain the following result in which denotes the isotype of the trivial representation. Let be a nontrivial irreducible representation of Then the isotype to the trivial representation of is the null space. That means the following equation holds Let be an orthonormal basis of Then we have: Therefore, the following is valid for a nontrivial irreducible representation : Example. Let be the permutation groups in three elements. Let be a linear representation of defined on the generating elements as follows: This representation can be decomposed on first look into the left-regular representation of which is denoted by in the following, and the representation with With the help of the irreducibility criterion taken from the next chapter, we could realize that is irreducible but is not. This is because (in terms of the inner product from ”Inner product and characters” below) we have The subspace of is invariant with respect to the left-regular representation. Restricted to this subspace we obtain the trivial representation. The orthogonal complement of is Restricted to this subspace, which is also –invariant as we have seen above, we obtain the representation given by Again, we can use the irreducibility criterion of the next chapter to prove that is irreducible. Now, and are isomorphic because for all in which is given by the matrix A decomposition of in irreducible subrepresentations is: where denotes the trivial representation and