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Let $G$ be a group that acts on a finite set $\Omega$. Let $V_{\Omega}$ be a $\mathbb{C}$-vector space whose basis elements $e_x$ are indexed by the elements $x$ of $\Omega$. In my book, it says that then we get a representation $\phi$ of $G$ on $V_{\Omega}$ by $$\phi(g) \sum_{x \in \Omega} a_xe_x = \sum_{x \in \Omega} a_xe_{g.x}$$

($a_x \in \mathbb{C}$). Furthermore, it is said that this type of representation is called a permutation representation.

Can someone explain this? I don't know what I am missing, but can someone explain this more detailed? Do I understand it correct that the homomorphism $G \to \text{GL}(n, \mathbb{C})$ is given by $g \mapsto BA^{-1}$ where $A$ is the matrix corresponding to the element $\sum_{x \in \Omega} a_xe_x$ and the same for $B$? I feel that I have understood this wrong.

Roger
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There is a group homomorphism $\psi:Bij(\Omega) \to GL(V_\Omega)$ (where $Bij(\Omega)$ is the group of bijection of $\Omega$) given by $\psi(f)(e_x)=e_{f(x)}$ extended by linearity. $\psi(f)$ is in $GL(V_\Omega)$ because it has inverse $\psi(f^{-1})$.

Now, a $G$-action on $\Omega$ is just a map $G \to Bij(\Omega)$. It is then trivial to see that a $G$-action on $\Omega$ induces a group representation structure on $V_\Omega$, just consider the composite group homomorphism $$G \to Bij(\Omega) \to GL(V_\Omega).$$

Nitrogen
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  • Thanks for your answer. Can you explain the part about the inverse? I didn't quite get it. Also, what do you mean by a $G$-action is just a map $G \to Bij(\Omega)$? I agree its a maps $G \times \Omega \to \Omega$, but what do you mean? – Roger Feb 14 '16 at 00:08
  • $GL(V_\Omega)$ is the group of invertible endomorphisms of $V_\Omega$, thus to show that $\psi(f)\in GL(V_\Omega)$ it is sufficient to show that it has an inverse, namely $\psi(f^{-1})$. For the $G$-action, they are usually defined as maps $u:G\times \Omega \to \Omega$ satisfying certain conditions. But these conditions imply that $\bar{u}:G \to Bij(\Omega)$ where $(\bar{u}(g))(s)=g\cdot s$ is a group homomorphism. – Nitrogen Feb 14 '16 at 00:18