Group Action of Symmetric Group on Complex Vector Space/Stabilizer/Examples/Example 2

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
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.