permutation representation
Let $G$ be a group, and $S$ any finite set on which $G$ acts.
That means that for any $g,h\in G$; $\mathbf{v},\mathbf{w}\in S$

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$g\mathbf{v}\in V$,

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$(gh)\mathbf{v}=g(h\mathbf{v})$,

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$e\mathbf{v}=\mathbf{v}$.
Notice that we almost have what it takes to make $S$ a representation^{} of $G$, but $S$ is no vector space^{}. We can however obtain a $G$module (a vector space carrying a representation of $G$) as follows.
Let $S=\{{\mathbf{s}}_{1},{\mathbf{s}}_{2},\mathrm{\dots},{\mathbf{s}}_{n}\}$. And let $\u2102S=\u2102[{\mathbf{s}}_{1},{\mathbf{s}}_{2},\mathrm{\dots},{\mathbf{s}}_{n}]$ be the vector space generated by $S$ over $\u2102$. in other words, $\u2102S$ is made of all formal linear combinations^{} ${c}_{1}{\mathbf{s}}_{1}+{c}_{2}{\mathbf{s}}_{2}+\mathrm{\cdots}+{c}_{n}{\mathbf{s}}_{n}$ with ${c}_{j}\in \u2102$. The sum is defined coordinatewise as is scalar multiplication.
Then the action of $G$ in $S$ can be extended linearly to $\u2102S$ as
$$g({c}_{1}{\mathbf{s}}_{1}+{c}_{2}{\mathbf{s}}_{2}+\mathrm{\cdots}+{c}_{n}{\mathbf{s}}_{n})={c}_{1}(g{\mathbf{s}}_{1})+{c}_{2}(g{\mathbf{s}}_{2})+\mathrm{\cdots}+{c}_{n}(g{\mathbf{s}}_{n})$$ 
and then the map $\rho :G\to GL(\u2102S)$ where $\rho $ is such that $\rho (g)(\mathbf{v})=g\mathbf{v}$ makes $\u2102S$ into a $G$module. The $G$module $\u2102S$ is known as the permutation representation associated with $S$.
Example.
If $G={S}_{n}$ acts on $S=\{\mathrm{\U0001d7cf},\mathrm{\U0001d7d0},\mathrm{\dots},\mathbf{n}\}$, then
$$\u2102S=\{{c}_{1}\mathrm{\U0001d7cf}+{c}_{2}\mathrm{\U0001d7d0}+\mathrm{\cdots}+{c}_{n}\mathbf{n}\}.$$ 
If $\sigma \in {S}_{n}$, the action becomes
$$\sigma ({c}_{1}\mathrm{\U0001d7cf}+{c}_{2}\mathrm{\U0001d7d0}+\mathrm{\cdots}+{c}_{n}\mathbf{n})={c}_{1}\sigma (\mathrm{\U0001d7cf})+{c}_{2}\sigma (\mathrm{\U0001d7d0})+\mathrm{\cdots}+{c}_{n}\sigma (\mathbf{n}).$$ 
Since $S$ forms a basis for this space, we can compute the matrices corresponding to the defining permutation^{} and we will see that the corresponding permutation matrices^{} are obtained.
References. Bruce E. Sagan. The Symmetric Group^{}: Representations, Combinatorial Algorithms and Symmetric Functions. 2a Ed. 2000. Graduate Texts in Mathematics. Springer.
Title  permutation representation 

Canonical name  PermutationRepresentation 
Date of creation  20130322 14:53:59 
Last modified on  20130322 14:53:59 
Owner  drini (3) 
Last modified by  drini (3) 
Numerical id  6 
Author  drini (3) 
Entry type  Definition 
Classification  msc 20Cxx 
Related topic  MatrixRepresentation 