# Definition:Linear Representation/Group

## Definition

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 $\rho: G \to \GL V$.

### Module associated to representation

Let $K \sqbrk G$ be the group ring.

Let $\map {\operatorname{End} } V$ be the endomorphism ring of $V$.

Let $K \sqbrk G \to \map {\operatorname{End} } V$ be the ring homomorphism given by $\rho : G \to \GL V$ and the Universal Property of Group Ring.

The $K \sqbrk G$-module induced by the representation is the module induced by this homomorphism.

## Also defined as

While a linear representation is sometimes defined as a linear group action, on $\mathsf{Pr} \infty \mathsf{fWiki}$ we make the distinction and define it as a permutation representation.