group representation
Let $G$ be a group, and let $V$ be a vector space^{}. A representation of $G$ in $V$ is a group homomorphism^{} $\rho :G\to \mathrm{GL}(V)$ from $G$ to the general linear group^{} $\mathrm{GL}(V)$ of invertible linear transformations of $V$.
Equivalently, a representation of $G$ is a vector space $V$ which is a $G$module, that is, a (left) module over the group ring^{} $\mathbb{Z}[G]$. The equivalence is achieved by assigning to each homomorphism^{} $\rho :G\to \mathrm{GL}(V)$ the module structure^{} whose scalar multiplication is defined by $g\cdot v:=(\rho (g))(v)$, and extending linearly. Note that, although technically a group representation^{} is a homomorphism such as $\rho $, most authors invariably denote a representation using the underlying vector space $V$, with the homomorphism being understood from context, in much the same way that vector spaces themselves are usually described as sets with the corresponding binary operations^{} being understood from context.
Special kinds of representations
(preserving all notation from above)
A representation is faithful if either of the following equivalent^{} conditions is satisfied:

•
$\rho :G\to \mathrm{GL}(V)$ is injective^{},

•
$V$ is a faithful left $\mathbb{Z}[G]$–module.
A subrepresentation of $V$ is a subspace^{} $W$ of $V$ which is a left $\mathbb{Z}[G]$–submodule^{} of $V$; such a subspace is sometimes called a $G$invariant subspace of $V$. Equivalently, a subrepresentation of $V$ is a subspace $W$ of $V$ with the property that
$$(\rho (g))(w)\in W\text{for all}g\in G\text{and}w\in W.$$ 
A representation $V$ is called irreducible if it has no subrepresentations other than itself and the zero module^{}.
Title  group representation 

Canonical name  GroupRepresentation 
Date of creation  20130322 12:13:30 
Last modified on  20130322 12:13:30 
Owner  djao (24) 
Last modified by  djao (24) 
Numerical id  9 
Author  djao (24) 
Entry type  Definition 
Classification  msc 20C99 
Synonym  representation 
Related topic  GeneralLinearGroup 
Defines  subrepresentation 
Defines  irreducible 
Defines  faithful 