Let $G$ a finite group which act on a finite set $M$. Let $C(M) : = Map(M, \mathbb{C}:= \{f: M \rightarrow \mathbb{C} \}$ the vector space of complex values functions from $M$. The group $G$ act on $C(M)$ via \begin{align} G \times C(M) \rightarrow C(M), \, \, (g, f) \mapsto g(f): M \rightarrow \mathbb{C}, \, \, m \mapsto f(g^{-1}(m)) \end{align} Show that $C(M)$ is a representation of $G$ and for the character $\chi_{C(M)}$ we have that $\chi_{C(M)}(g) = |M^{G}| \, \, \, \forall \, g\in G$, where $M^{G}:= \{m\in M | g(m) = m \, \, \, \forall g\in G \}$ is the set of fixed points in $M$ for the $G$-action.

I showed that $C(M)$ is a representation of $G$, but I have some problem with the proof of $$\chi_{C(M)}(g) = |M^{G}| \, \, \, \forall \, g\in G$$ Any suggestions? Thanks in advance!

  • $\begingroup$ An element $g$ acts by some matrix, and $\chi(g)$ is the trace of that matrix. Have you tried any examples? $\endgroup$ – Joppy Feb 27 at 3:11
  • $\begingroup$ @Joppy No, I tried to compute the associated matrix but I had no idea how to construct a basis of $C(M)$ $\endgroup$ – userr777 Feb 27 at 22:16
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    $\begingroup$ A pretty usual choice is to use the indicator functions: for each $m \in M$, let $\delta_m: M \to \mathbb{C}$ which takes the value $1$ at $m$, and $0$ at any other element of $M$. $\endgroup$ – Joppy Feb 28 at 2:04

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