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 $V$.
$\Box$
$\blacksquare$
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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.
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Therefore:
- $\hat{}$ : (linear representations) $\to$ (linear actions)
- $\tilde{}$ : (linear actions) $\to$ (linear representations)
give a bijection.
$\blacksquare$