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Definitions

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The statement of the theorem

A representation of a groupG on a vector spaceV over a fieldK is a group homomorphism from G to GL(V), the general linear group on V. That is, a representation is a map ρ ⁣:G→GL(V)\rho \colon G\to \mathrm {GL} \left(V\right) such that ρ(g1g2)=ρ(g1)ρ(g2),for all g1,g2∈G.\rho (g_{1}g_{2})=\rho (g_{1})\rho (g_{2}),\qquad {\text{for all }}g_{1},g_{2}\in G. Here V is called the representation space and the dimension of V is called the dimension or degree of the representation. It is common practice to refer to V itself as the representation when the homomorphism is clear from the context. In the case where V is of finite dimension n it is common to choose a basis for V and identify GL(V) with GL(n, K), the group of n×nn\times n invertible matrices on the field K. - If G is a topological group and V is a topological vector space, a continuous representation of G on V is a representation ρ such that the application Φ : G × V → V defined by Φ(g, v) = ρ(g)(v) is continuous. - The kernel of a representation ρ of a group G is defined as the normal subgroup of G whose image under ρ is the identity transformation: ker⁡ρ={g∈G∣ρ(g)=id}.\ker \rho =\left\{g\in G\mid \rho (g)=\mathrm {id} \right\}. A faithful representation is one in which the homomorphism G → GL(V) is injective; in other words, one whose kernel is the trivial subgroup {e} consisting only of the group's identity element. - Given two K vector spaces V and W, a map of two representations ρ : G → GL(V) and π : G → GL(W) (also called an equivariant map) is a linear map α : V → W such that α∘ρ(g)=π(g)∘α.\alpha \circ \rho (g)=\pi (g)\circ \alpha . If α has an inverse equivariant map, the map is called an isomorphism of representations and the representations are said to be equivalent or isomorphic. It is also sufficient that α is an equivariant map which is an isomorphism of vector spaces.