Group Action of Symmetric Group on Complex Vector Space/Orbit/Examples/Example 1

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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 $v = v_1 + v_2 + v_3 + v_4$.

The orbit of $v$ is:

$\Orb v = \set v$


Proof

\(\ds \Orb v\) \(=\) \(\ds \set {w \in V: \exists \rho \in S_4: w = \rho * v}\) Definition of Orbit
\(\ds \) \(=\) \(\ds \set {w \in V: \exists \rho \in S_4: w = \rho * \sum_{k \mathop = 1}^4 v_k}\) Definition of $v$
\(\ds \) \(=\) \(\ds \set {w \in V: \exists \rho \in S_4: w = \sum_{k \mathop = 1}^4 v_\map \rho k}\) Definition of $*$
\(\ds \) \(=\) \(\ds \set {w \in V: \exists \rho \in S_4: w = \sum_{k \mathop = 1}^4 v_k}\) Permutation of Indices of Summation
\(\ds \) \(=\) \(\ds \set {w \in V: \exists \rho \in S_4: w = v}\) Definition of $v$
\(\ds \) \(=\) \(\ds \set v\)

$\blacksquare$


Sources

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