Group Action of Symmetric Group on Complex Vector Space/Stabilizer/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 stabilizer of $v$ is:

$\Stab v = \set {e, \tuple {1 3}, \tuple {2 4}, \tuple {1 3} \tuple {2 4} }$


Proof

\(\ds \Stab v\) \(=\) \(\ds \set {\rho \in S_4: \rho * v = v}\) Definition of Stabilizer
\(\ds \) \(=\) \(\ds \set {\rho \in S_4: \rho * \paren {v_1 + v_3} = v_1 + v_3}\) Definition of $v$
\(\ds \) \(=\) \(\ds \set {\rho \in S_4: v_{\map \rho 1} + v_{\map \rho 3} = v_1 + v_3}\) Definition of $*$
\(\ds \) \(=\) \(\ds \set {\rho \in S_4: \set {\map \rho 1, \map \rho 3} = \set {1, 3} }\) simplifying


Thus $\Stab v$ consists of all the permutations of $S_4$ which either:

fix $1$ and $3$

or:

transpose $1$ and $3$.


We have that:

$e$ and $\tuple {2 4}$ are the permutations which fix $1$ and $3$
$\tuple {1 3}$ and $\tuple {1 3} \tuple {2 4}$ are the permutations which transpose $1$ and $3$.

Hence the result.

$\blacksquare$


Sources

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