This definition can be iterated a finite number of times. Let and be representations of the group Then is a representation by virtue of the following identity: . Let and let be the representation on Let be the representation on and the representation on Then the identity above leads to the following result: for all Theorem. The irreducible representations of up to isomorphism are exactly the representations in which and are irreducible representations of and respectively. Symmetric and alternating square Let be a linear representation of Let be a basis of Define by extending linearly. It then holds that and therefore splits up into in which These subspaces are –invariant and by this define subrepresentations which are called the symmetric square and the alternating square, respectively. These subrepresentations are also defined in although in this case they are denoted wedge product and symmetric product In case that the vector space is in general not equal to the direct sum of these two products. Decompositions In order to understand representations more easily, a decomposition of the representation space into the direct sum of simpler subrepresentations would be desirable. This can be achieved for finite groups as we will see in the following results. More detailed explanations and proofs may be found in [1] and [2]. Theorem. (Maschke) Let be a linear representation where is a vector space over a field of characteristic zero. Let be a -invariant subspace of Then the complement of exists in and is -invariant. A subrepresentation and its complement determine a representation uniquely. The following theorem will be presented in a more general way, as it provides a very beautiful result about representations of compact – and therefore also of finite – groups: Theorem. Every linear representation of a compact group over a field of characteristic zero is a direct sum of irreducible representations. Or in the language of -modules: If the group algebra is semisimple, i.e. it is the direct sum of simple algebras. Note that this decomposition is not unique. However, the number of how many times a subrepresentation isomorphic to a given irreducible representation is occurring in this decomposition is independent of the choice of decomposition. The canonical decomposition To achieve a unique decomposition, one has to combine all the irreducible subrepresentations that are isomorphic to each other. That means, the representation space is decomposed into a direct sum of its isotypes. This decomposition