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
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $10$: The Orbit-Stabiliser Theorem: Exercise $1$