SQUARE45

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

\rho \colon G\to \mathrm {GL} \left(V\right)

such that

\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\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 \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

\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.

Group Representations

Sequence of Expressions

Definitions

Maschke's Theorem