Group Action of Symmetric Group on Complex Vector Space

From ProofWiki
Jump to navigation Jump to search

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:

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


Then $*$ is a group action.


Orbit of Element of $V$

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

$\ds \Orb v = \set {w \in V: \exists \rho \in S_n: w = \sum_{k \mathop = 1}^n \lambda_k v_{\map \rho k} }$


Stabilizer of Element of $V$

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

Let $\rho, \sigma \in S_n$.

Let $v = \ds \sum_{k \mathop = 1}^n \lambda_k v_k$.


We have:

\(\ds \rho * \paren {\sigma * v}\) \(=\) \(\ds \rho * \paren {\sigma * \sum_{k \mathop = 1}^n \lambda_k v_k}\) Definition of $v$
\(\ds \) \(=\) \(\ds \rho * \sum_{k \mathop = 1}^n \lambda_k v_{\map \sigma k}\) Definition of $*$
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^n \lambda_k v_{\map \rho {\map \sigma k} }\) Definition of $*$
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^n \lambda_k v_{\map {\rho \mathop \circ \sigma} k}\) Definition of Composition of Mappings
\(\ds \) \(=\) \(\ds \paren {\rho \circ \sigma} * \sum_{k \mathop = 1}^n \lambda_k v_k\) Definition of $*$
\(\ds \) \(=\) \(\ds \paren {\rho \circ \sigma} * v\) Definition of $v$

Hence $*$ fulfils Group Action Axiom $\text {GA} 1$.


Let $e$ denote the identity element of $S_n$. Then:

\(\ds e * v\) \(=\) \(\ds e * \sum_{k \mathop = 1}^n \lambda_k v_k\) Definition of $v$
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^n \lambda_k v_{\map e k}\) Definition of $*$
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^n \lambda_k v_k\) Definition of $e$
\(\ds \) \(=\) \(\ds v\) Definition of $v$

Hence $*$ fulfils Group Action Axiom $\text {GA} 2$.

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Group_Action_of_Symmetric_Group_on_Complex_Vector_Space&oldid=541385"