# group representation

Let $G$ be a group, and let $V$ be a vector space. A representation of $G$ in $V$ is a group homomorphism $\rho\colon G\to\operatorname{GL}(V)$ from $G$ to the general linear group $\operatorname{GL}(V)$ 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 $\rho\colon G\to\operatorname{GL}(V)$ 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:

• $\rho\colon G\to\operatorname{GL}(V)$ 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 GroupRepresentation 2013-03-22 12:13:30 2013-03-22 12:13:30 djao (24) djao (24) 9 djao (24) Definition msc 20C99 representation GeneralLinearGroup subrepresentation irreducible faithful