Editing Group representation (section)

==Definitions==
A '''representation''' of a [[group (mathematics)|group]] ''G'' on a [[vector space]] ''V'' over a [[field (mathematics)|field]] ''K'' is a [[group homomorphism]] from ''G'' to GL(''V''), the [[general linear group#General linear group of a vector space|general linear group]] on ''V''.  That is, a representation is a map 
:<math>\rho \colon G \to \mathrm{GL}\left(V \right)</math> 
such that
:<math>\rho(g_1 g_2) = \rho(g_1) \rho(g_2) , \qquad \text{for all }g_1,g_2 \in G.</math>

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 (linear algebra)|basis]] for ''V'' and identify GL(''V'') with {{nowrap|GL(''n'', ''K'')}}, the group of <math>n \times n</math> [[invertible matrix|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 {{nowrap|Φ : ''G'' × ''V'' → ''V''}} defined by {{nowrap|1=Φ(''g'', ''v'') = ''ρ''(''g'')(''v'')}} is [[continuous function (topology)|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:
::<math>\ker \rho = \left\{g \in G \mid \rho(g) = \mathrm{id}\right\}.</math>

: A [[faithful representation]] is one in which the homomorphism {{nowrap|''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'', two representations {{nowrap|''ρ'' : ''G'' → GL(''V'')}} and {{nowrap|''π'' : ''G'' → GL(''W'')}} are said to be '''equivalent''' or '''isomorphic''' if there exists a vector space [[isomorphism]] {{nowrap|''α'' : ''V'' → ''W''}} so that for all ''g'' in ''G'',
::<math>\alpha \circ \rho(g) \circ \alpha^{-1} = \pi(g).</math>