definition is consistent with the definition of the Fourier transformation on Let be a representation and let be a -valued function on . The Fourier transform of is defined as This transformation satisfies Maps between representations A map between two representations of the same group is a linear map with the property that holds for all In other words, the following diagram commutes for all : Such a map is also called –linear, or an equivariant map. The kernel, the image and the cokernel of are defined by default. The composition of equivariant maps is again an equivariant map. There is a category of representations with equivariant maps as its morphisms. They are again –modules. Thus, they provide representations of due to the correlation described in the previous section. Irreducible representations and Schur's lemma Let be a linear representation of Let be a -invariant subspace of that is, for all and . The restriction is an isomorphism of onto itself. Because holds for all this construction is a representation of in It is called subrepresentation of Any representation V has at least two subrepresentations, namely the one consisting only of 0, and the one consisting of V itself. The representation is called an irreducible representation, if these two are the only subrepresentations. Some authors also call these representations simple, given that they are precisely the simple modules over the group algebra . Schur's lemma puts a strong constraint on maps between irreducible representations. If and are both irreducible, and is a linear map such that for all , there is the following dichotomy: If and is a homothety (i.e. for a ). More generally, if and are isomorphic, the space of G-linear maps is one-dimensional. Otherwise, if the two representations are not isomorphic, F must be 0. Properties Two representations are called equivalent or isomorphic, if there exists a –linear vector space isomorphism between the representation spaces. In other words, they are isomorphic if there exists a bijective linear map such that for all In particular, equivalent representations have the same degree. A representation is called faithful when is injective. In this case induces an isomorphism between and the image As the latter is a subgroup of we can regard via as subgroup of We can restrict the range as well as the domain: Let be a subgroup of Let be a linear representation of We denote by the restriction