Let be any vector space isomorphism. Then which is defined by the equation for all is a representation isomorphic to By restricting the domain of the representation to a subgroup, e.g. we obtain the representation This representation is defined by the image whose explicit form is shown above. Constructions The dual representation Let be a given representation. The dual representation or contragredient representation is a representation of in the dual vector space of It is defined by the property With regard to the natural pairing between and the definition above provides the equation: For an example, see the main page on this topic: Dual representation. Direct sum of representations Let and be a representation of and respectively. The direct sum of these representations is a linear representation and is defined as Let be representations of the same group For the sake of simplicity, the direct sum of these representations is defined as a representation of i.e. it is given as by viewing as the diagonal subgroup of Example. Let (here and are the imaginary unit and the primitive cube root of unity respectively): Then As it is sufficient to consider the image of the generating element, we find that Tensor product of representations Let be linear representations. We define the linear representation into the tensor product of and by in which This representation is called outer tensor product of the representations and The existence and uniqueness is a consequence of the properties of the tensor product. Example. We reexamine the example provided for the direct sum: The outer tensor product Using the standard basis of we have the following for the generating element: Remark. Note that the direct sum and the tensor products have different degrees and hence are different representations. Let be two linear representations of the same group. Let be an element of Then is defined by for and we write Then the map defines a linear representation of which is also called tensor product of the given representations. These two cases have to be strictly distinguished. The first case is a representation of the group product into the tensor product of the corresponding representation spaces. The second case is a representation of the group into the tensor product of two representation spaces of this one group. But this last case can be viewed as a special case of the first one by focusing on the diagonal subgroup