Correspondence between Linear Group Actions and Linear Representations

Theorem
Let $\struct {k, +, \cdot}$ be a field.

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

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

There is a one-to-one correspondence between linear group actions of $G$ on $V$ and linear representations of $G$ in $V$, as follows:

Let $\phi: G \times V \to V$ be a group action.

Let $\rho: G \to \map {\operatorname {Sym} } V$ be a permutation representation of $G$ on $V$.

The following are equivalent:


 * $(1): \quad$ $\rho$ is the permutation representation associated to $\phi$


 * $(2): \quad$ $\phi$ is the group action associated to $\rho$

If this is the case, the following are equivalent:


 * $(1): \quad$ $\rho$ is a linear representation


 * $(2): \quad$ $\phi$ is a linear group action

Proof
The first equivalence follows from Correspondence Between Group Actions and Permutation Representations.

1 implies 2
Let $\rho : G \to \GL V$ be a linear representation of $G$ on $V$.

Then for all $g \in G$, $v_1, v_2 \in V$:

and for all $g \in G$, $v \in V$, $\lambda \in k$:

Therefore $\phi$ is a linear group action of $G$ on $V$.

2 implies 1
Let $\phi: G \times V \to V$ be a linear action of $G$ on $V$.

Then for all $g \in G$, $v_1, v_2 \in V$:

and for all $g \in G$, $v \in V$, $\lambda \in k$:

Therefore $\rho$ is a linear representation of $G$ on $V$.

Now let $g_1, g_2 \in G$.

We have for all $v \in V$:

Thus $\rho$ satisfies the homomorphism property.

Therefore:


 * $\hat{}$ : (linear representations) $\to$ (linear actions)
 * $\tilde{}$ : (linear actions) $\to$ (linear representations)

give a bijection.

Also see

 * Correspondence Between Group Actions and Permutation Representations