Group Representation - Definitions

Definitions

A representation of a group G on a vector space V over a field K is a group homomorphism from G to GL(V), the general linear group on V. That is, a representation is a map

such that

Here V is called the representation space and the dimension of V is called the dimension 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-by-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 defined by 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:

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 of just the group's identity element.

• Given two K vector spaces V and W, two representations

and

are said to be equivalent or isomorphic if there exists a vector space isomorphism

so that for all g in G