Definition:Linear Representation

Definition

Groups

Let $\struct {K, +, \circ}$ be a field.

Let $V$ be a vector space over $K$ of finite dimension.

Let $\GL V$ be the general linear group of $V$.

Let $\struct {G, \cdot}$ be a finite group.

A linear representation of $G$ on $V$ is a group homomorphism $\rho: G \to \GL V$.

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:

$\rho: A \to \map {\operatorname {End} } V$

where $\map {\operatorname {End} } V$ is the endomorphism ring of $V$.

Also see

• Definition:Linear Group Action
• Definition:Permutation Representation