Group Action of Symmetric Group on Complex Vector Space/Orbit/Examples/Example 2
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Example of Orbit of Group Action of Symmetric Group on Complex Vector Space
Let $S_n$ denote the symmetric group on $n$ letters.
Let $V$ denote a vector space over the complex numbers $\C$.
Let $V$ have a basis:
- $\BB := \set {v_1, v_2, \ldots, v_n}$
Let $*: S_n \times V \to V$ be a group action of $S_n$ on $V$ defined as:
- $\forall \tuple {\rho, v} \in S_n \times V: \rho * v := \lambda_1 v_{\map \rho 1} + \lambda_2 v_{\map \rho 2} + \dotsb + \lambda_n v_{\map \rho n}$
where:
- $v = \lambda_1 v_1 + \lambda_2 v_2 + \dotsb + \lambda_n v_n$
Let $n = 4$.
Let $v = v_1 + v_3$.
The orbit of $v$ is:
- $\Orb v = \set {v_1 + v_2, v_1 + v_3, v_1 + v_4, v_2 + v_3, v_2 + v_4, v_3 + v_4}$
Proof
\(\ds \Orb v\) | \(=\) | \(\ds \set {w \in V: \exists \rho \in S_4: w = \rho * v}\) | Definition of Orbit | |||||||||||
\(\ds \) | \(=\) | \(\ds \set {w \in V: \exists \rho \in S_4: w = \rho * \paren {v_1 + v_3} }\) | Definition of $v$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \set {w \in V: \exists \rho \in S_4: w = v_{\map \rho 1} + v_{\map \rho 3} }\) | Definition of $*$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \set {v_1 + v_2, v_1 + v_3, v_1 + v_4, v_2 + v_1, v_2 + v_3, v_2 + v_4, v_3 + v_1, v_3 + v_2, v_3 + v_4, v_4 + v_1, v_4 + v_2, v_4 + v_3}\) | all permutations of $2$ from $4$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \set {v_1 + v_2, v_1 + v_3, v_1 + v_4, v_2 + v_3, v_2 + v_4, v_3 + v_4}\) | removing duplicates, as $\Orb v$ is a set |
$\blacksquare$
Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $10$: The Orbit-Stabiliser Theorem: Exercise $1 \ \text {(b)}$