Correspondence between Linear Group Actions and Linear Representations

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Theorem

Let $\left({k, +, \cdot}\right)$ be a field.

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

Let $\left({G, *}\right)$ 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 $\rho : G \to \operatorname{Sym}(V)$ 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 $\rho : G \to \operatorname{GL} \left({V}\right)$ be a linear representation of $G$ on $V$.

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

\(\displaystyle \phi \left({g, v_1 + v_2}\right)\) \(=\) \(\displaystyle \rho \left({g}\right) \left({v_1 + v_2}\right)\) $\quad$ Definition of Group Action Associated to Permutation Representation $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \rho \left({g}\right) \left({v_1}\right) + \rho \left({g}\right) \left({v_2}\right)\) $\quad$ $\rho \left({g}\right)$ is linear $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \phi \left({g, v_1}\right) + \phi \left({g, v_2}\right)\) $\quad$ Definition of Group Action Associated to Permutation Representation $\quad$

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

\(\displaystyle \phi \left({g, \lambda \cdot v }\right)\) \(=\) \(\displaystyle \rho \left({g}\right) \left({\lambda \cdot v}\right)\) $\quad$ Definition of Group Action Associated to Permutation Representation $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \lambda \cdot \rho\left({g}\right) \left({v}\right)\) $\quad$ $\rho \left({g}\right)$ is linear $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \lambda \cdot \phi \left({g, v}\right)\) $\quad$ Definition of Group Action Associated to Permutation Representation $\quad$

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

\(\displaystyle \rho \left({g}\right) \left({v_1 + v_2}\right)\) \(=\) \(\displaystyle \phi \left({g, v_1 + v_2}\right)\) $\quad$ Definition of Permutation Representation Associated to Group Action $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \phi \left({g, v_1}\right) + \phi \left({g, v_2}\right)\) $\quad$ Definition of Linear Group Action $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \rho \left({g}\right) \left({v_1}\right) + \rho \left({g}\right) \left({v_2}\right)\) $\quad$ Definition of Permutation Representation Associated to Group Action $\quad$

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

\(\displaystyle \rho \left({g}\right) \left({\lambda \cdot v}\right)\) \(=\) \(\displaystyle \phi \left({g, \lambda \cdot v }\right)\) $\quad$ Definition of Permutation Representation Associated to Group Action $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \lambda \cdot \phi \left({g, v}\right)\) $\quad$ Definition of Linear Group Action $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \lambda \cdot \rho\left({g}\right) \left({v}\right)\) $\quad$ Definition of Permutation Representation Associated to Group Action $\quad$

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

$\Box$

$\blacksquare$


Now let $g_1, g_2 \in G$.

We have for all $v \in V$:

\(\displaystyle \rho \left({g_1 * g_2}\right) \left({v}\right)\) \(=\) \(\displaystyle \phi \left({g_1 * g_2, v}\right)\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \phi \left({g_1, \phi\left({g_2, v}\right) }\right)\) $\quad$ $\phi$ is an action $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \rho \left({g_2}\right) \left({\rho \left({g_1}\right) \left({v}\right) }\right)\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left({ \rho \left({g_2}\right) \circ \rho \left({g_1}\right) }\right) \left({v}\right)\) $\quad$ where $\circ$ is the composition of mappings $\quad$

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