# Definition:Linear Representation

## Definition

### Groups

Let $\left({\mathbb k, +, \circ}\right)$ be a field.

Let $V$ be a vector space over $\mathbb k$ of finite dimension.

Let $\operatorname {GL} \left({V}\right)$ be the general linear group of $V$.

Let $\left({G, \cdot}\right)$ be a finite group.

A linear representation of $G$ on $V$ is a group homomorphism $\rho: G \to \operatorname {GL} \left({V}\right)$.

### Algebras

Let $K$ be a field.

Let $A$ be an associative unitary algebra over $K$.

Then a (linear) representation of $A$ is a vector space $V$ over $K$ equipped with a homomorphism of algebras:

$\rho: A \to \operatorname{End} \left({V}\right)$

where $\operatorname{End}\left({V}\right)$ is the endomorphism ring of $V$.