Correspondence between Linear Group Actions and Linear Representations

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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.

$\blacksquare$


Also see