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 $S=\\{\\mathbf{s}_{1},\\mathbf{s}_{2},\\ldots,\\mathbf{s}_{n}\\}$ . And let $\\mathbbmss{C}S=\\mathbbmss{C}[\\mathbf{s}_{1},\\mathbf{s}_{2},\\ldots,\\mathbf{s}_{%n}]$ be the vector space generated by $S$ over $\\mathbbmss{C}$ . in other words, $\\mathbbmss{C}S$ is made of all formal linear combinations $c_{1}\\mathbf{s}_{1}+c_{2}\\mathbf{s}_{2}+\\cdots+c_{n}\\mathbf{s}_{n}$ with $c_{j}\\in\\mathbbmss{C}$ .The sum is defined coordinate-wise as is scalar multiplication.

Then the action of $G$ in $S$ can be extended linearly to $\\mathbbmss{C}S$ as

g(c_{1}\\mathbf{s}_{1}+c_{2}\\mathbf{s}_{2}+\\cdots+c_{n}\\mathbf{s}_{n})=c_{1}(g%\\mathbf{s}_{1})+c_{2}(g\\mathbf{s}_{2})+\\cdots+c_{n}(g\\mathbf{s}_{n})

and then the map $\\rho:G\\to GL(\\mathbbmss{C}S)$ where $\\rho$ is such that $\\rho(g)(\\mathbf{v})=g\\mathbf{v}$ makes $\\mathbbmss{C}S$ into a $G$ -module. The $G$ -module $\\mathbbmss{C}S$ is known as the permutation representation associated with $S$ .

Example.
If $G=S_{n}$ acts on $S=\\{\\mathbf{1},\\mathbf{2},\\ldots,\\mathbf{n}\\}$ , then

\\mathbbmss{C}S=\\{c_{1}\\mathbf{1}+c_{2}\\mathbf{2}+\\cdots+c_{n}\\mathbf{n}\\}.

If $\\sigma\\in S_{n}$ , the action becomes

\\sigma(c_{1}\\mathbf{1}+c_{2}\\mathbf{2}+\\cdots+c_{n}\\mathbf{n})=c_{1}\\sigma(%\\mathbf{1})+c_{2}\\sigma(\\mathbf{2})+\\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.