# 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 $g \in G$, $v_1, v_2 \in V$:

\(\ds \map \phi {g, v_1 + v_2}\) | \(=\) | \(\ds \map {\map \rho g} {v_1 + v_2}\) | Definition of Group Action Associated to Permutation Representation | |||||||||||

\(\ds \) | \(=\) | \(\ds \map {\map \rho g} {v_1} + \map {\map \rho g} {v_2}\) | $\map \rho g$ is linear | |||||||||||

\(\ds \) | \(=\) | \(\ds \map \phi {g, v_1} + \map \phi {g, v_2}\) | Definition of Group Action Associated to Permutation Representation |

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

\(\ds \map \phi {g, \lambda \cdot v}\) | \(=\) | \(\ds \map {\map \rho g} {\lambda \cdot v}\) | Definition of Group Action Associated to Permutation Representation | |||||||||||

\(\ds \) | \(=\) | \(\ds \map {\map {\lambda \cdot \rho} g} v\) | $\map \rho g$ is linear | |||||||||||

\(\ds \) | \(=\) | \(\ds \map {\lambda \cdot \phi} {g, v}\) | Definition of Group Action Associated to Permutation Representation |

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

$\Box$

### 2 implies 1

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

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

\(\ds \map {\map \rho g} {v_1 + v_2}\) | \(=\) | \(\ds \map \phi {g, v_1 + v_2}\) | Definition of Permutation Representation Associated to Group Action | |||||||||||

\(\ds \) | \(=\) | \(\ds \map \phi {g, v_1} + \map \phi {g, v_2}\) | Definition of Linear Group Action | |||||||||||

\(\ds \) | \(=\) | \(\ds \map {\map \rho g} {v_1} + \map {\map \rho g} {v_2}\) | Definition of Permutation Representation Associated to Group Action |

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

\(\ds \map {\map \rho g} {\lambda \cdot v}\) | \(=\) | \(\ds \map \phi {g, \lambda \cdot v}\) | Definition of Permutation Representation Associated to Group Action | |||||||||||

\(\ds \) | \(=\) | \(\ds \lambda \cdot \map \phi {g, v}\) | Definition of Linear Group Action | |||||||||||

\(\ds \) | \(=\) | \(\ds \map {\map {\lambda \cdot \rho} g} v\) | Definition of Permutation Representation Associated to Group Action |

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

$\Box$

$\blacksquare$

Now let $g_1, g_2 \in G$.

We have for all $v \in V$:

\(\ds \map {\map \rho {g_1 * g_2} } v\) | \(=\) | \(\ds \map \phi {g_1 * g_2, v}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \map \phi {g_1, \map \phi {g_2, v} }\) | $\phi$ is an action | |||||||||||

\(\ds \) | \(=\) | \(\ds \map {\map \rho {g_2} } {\map {\map \rho {g_1} } v}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \map {\paren {\map \rho {g_2} \circ \map \rho {g_1} } } v\) | where $\circ$ is the composition of mappings |

Thus $\rho$ satisfies the homomorphism property.

Therefore:

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

give a bijection.

$\blacksquare$