# Character of representation of map from a finite set $M$ to $\mathbb{C}$

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!

• An element $g$ acts by some matrix, and $\chi(g)$ is the trace of that matrix. Have you tried any examples? – Joppy Feb 27 at 3:11
• @Joppy No, I tried to compute the associated matrix but I had no idea how to construct a basis of $C(M)$ – userr777 Feb 27 at 22:16
• 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 $M$. – Joppy Feb 28 at 2:04