Group Action of Symmetric Group on Complex Vector Space/Stabilizer

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Theorem

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

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:

$v = \lambda_1 v_1 + \lambda_2 v_2 + \dotsb + \lambda_n v_n$


The stabilizer of an element $v \in V$ is:

$\ds \Stab v = \set {\rho \in S_n: \sum_{k \mathop = 1}^n \lambda_k v_k = \sum_{k \mathop = 1}^n \lambda_{\map \rho k} v_k}$


Proof

By definition:

\(\ds \Stab v\) \(=\) \(\ds \set {\rho \in S_n: \rho * v = v}\) Definition of Stabilizer
\(\ds \) \(=\) \(\ds \set {\rho \in S_n: \rho * \sum_{k \mathop = 1}^n \lambda_k v_k = \sum_{k \mathop = 1}^n \lambda_k v_k}\) Definition of $v$
\(\ds \) \(=\) \(\ds \set {\rho \in S_n: \sum_{k \mathop = 1}^n \lambda_k v_k = \sum_{k \mathop = 1}^n \lambda_{\map \rho k} v_k}\) Definition of $*$

$\blacksquare$


Examples

Example 1

Let $n = 4$.

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

The stabilizer of $v$ is:

$\Stab v = S_4$


Example 2

Let $n = 4$.

Let $v = v_1 + v_3$.

The stabilizer of $v$ is:

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


Sources