Let $G$ be a group, and $S$ any finite set on which $G$ acts.

That means that for any $g,h\in G$ ; $\vct{v},\vct{w}\in S$

• $g\vct{v}\in V$ ,
• $(gh)\vct v = g(h\vct v)$ ,
• $e\vct v = \vct v$ .

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 $S=\{\vct{s}_1,\vct{s}_2,\ldots,\vct{s}_n\}$ . And let be the vector space generated by $S$ over . in other words, is made of all formal linear combinations $c_1\vct{s}_1+c_2\vct{s}_2+\cdots+c_n\vct{s}_n$ with . The sum is defined coordinate-wise as is scalar multiplication.

Then the action of $G$ in $S$ can be extended linearly to as $$ g(c_1\vct{s}_1+c_2\vct{s}_2+\cdots+c_n\vct{s}_n)= c_1(g\vct{s}_1)+c_2(g\vct{s}_2)+\cdots+c_n(g\vct{s}_n) $$ and then the map where $\rho$ is such that $\rho(g)(\vct v) = g\vct v$ makes into a $G$ -module. The $G$ -module is known as the permutation representation associated with $S$ .

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

References. Bruce E. Sagan. The Symmetric Group: Representations, Combinatorial Algorithms and Symmetric Functions. 2a Ed. 2000. Graduate Texts in Mathematics. Springer.