Definition:Linear Representation/Group

Definition
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 $\rho: G \to \operatorname {GL} \left({V}\right)$.

Module associated to representation
Let $K \left[{G}\right]$ be the group ring.

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

Let $K \left[{G}\right] \to \operatorname{End}(V)$ be the ring homomorphism given by $\rho : G \to \operatorname {GL} \left({V}\right)$ and the Universal Property of Group Ring.

The $K \left[{G}\right]$-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 we make the distinction and define it as a permutation representation.

See also Existence of Bijection between Linear Group Action and Linear Representation.

Also see
By Existence of Bijection between Linear Group Action and Linear Representation, a linear representation of $G$ on $V$ is completely specified by a linear action of $G$ on $V$.