is the corresponding decomposition of the representation space. We obtain the canonical decomposition by combining all the isomorphic irreducible subrepresentations: is the -isotype of and consequently the canonical decomposition is given by The theorems above are in general not valid for infinite groups. This will be demonstrated by the following example: let Together with the matrix multiplication is an infinite group. acts on by matrix-vector multiplication. We consider the representation for all The subspace is a -invariant subspace. However, there exists no -invariant complement to this subspace. The assumption that such a complement exists would entail that every matrix is diagonalizable over This is known to be wrong and thus yields a contradiction. The moral of the story is that if we consider infinite groups, it is possible that a representation - even one that is not irreducible - can not be decomposed into a direct sum of irreducible subrepresentations. Character theory Definitions The character of a representation is defined as the map in which denotes the trace of the linear map Even though the character is a map between two groups, it is not in general a group homomorphism, as the following example shows. Let be the representation defined by: The character is given by Characters of permutation representations are particularly easy to compute. If V is the G-representation corresponding to the left action of on a finite set , then For example, the character of the regular representation is given by where denotes the neutral element of Properties A crucial property of characters is the formula This formula follows from the fact that the trace of a product AB of two square matrices is the same as the trace of BA. Functions satisfying such a formula are called class functions. Put differently, class functions and in particular characters are constant on each conjugacy class It also follows from elementary properties of the trace that is the sum of the eigenvalues of with multiplicity. If the degree of the representation is n, then the sum is n long. If s has order m, these eigenvalues are all m-th roots of unity. This fact can be used to show that and it also implies Since the trace of the identity matrix is the number of rows, where is the neutral element of and n is the dimension of the representation. In general, is a normal subgroup in The following table