I am reading Group Theory and Physics by Sternberg. On page 64, Section 2.5 Actions on function spaces, he states that:
$$ \text{dim} \space \text{Hom}_{G}(F(M),F(N))$$ is the number of orbits of $G$ in $N \times M$. I am asking why.
Notation: $G$ is a finite group. $M$ and $N$ are finite sets. $F(M)$ and $F(N)$ are the vector space of all complex-valued functions on $M$ and $N$, respectively.
The context is that we are given an action of $G$ on $F(M)$ and $F(N)$:
$$a f(x) = f(a^{-1} x) \quad \forall a \in G$$ $$a h(y) = h(a^{-1} y) \quad \forall a \in G$$
where $x \in M$, $f \in F(M)$, $y \in N$ and $h \in F(N)$.
$\text{Hom}(F(M),F(N))$ is the vector space of all linear maps from $F(M)$ to $F(N)$.
$$T_{K} \in \text{Hom}(F(M),F(N)) \implies (T_{K}f)(y) = \sum_{x \in M} K(y,x)f(x) \quad \forall (y,x) \in N \times M$$
where $f: M \rightarrow \mathbb{C}$, and $K: N \times M \rightarrow \mathbb{C}$ is a function we can associate with $T$; it describes what $T$ does to $f$. As the notation suggests, there is a bijective map $Z: F(N \times M) \rightarrow \text{Hom}(F(M),F(N))$, such that $Z(K) = T_K$. This map is a vector space isomorphism from $F(N \times M)$ to $\text{Hom}(F(M),F(N))$.
Finally, adding a subscript $G$ to $\text{Hom}$ indicates the subspace containing $T_K$ which are fixed by the action of the group.
$$\text{Hom}_{G}(F(M),F(N)) = \{T_K \in \text{Hom}(F(M),F(N)) \quad \vert \quad r^{\text{Hom}}(a)T_K = T_K \quad \forall a \in G \}$$
where $r^{\text{Hom}}$ is a representation of $G$ on $\text{Hom}(F(M),F(N))$.
Background and confusion: After proving that the isomorphism $Z$ is a $G$-morphism, Sternberg demonstrates that:
$$T_{K} \in \text{Hom}_{G}(F(M),F(N)) \implies K(a^{-1}y, a^{-1}x) = K(y,x) \quad \forall a \in G$$
meaning $K$ must be constant on the orbits of $G$ on $N \times M$. I think I understand (at least, follow) this.
What is not clear is how this determines the dimension of $\text{Hom}_{G}(F(M),F(N))$. There is obviously some one-to-one correspondence with $K$ being constant on the orbits of $N \times M$ and some analogous statement about the $T_K$, but I'm not making the connection.
This implication is actually a biconditional. The point is that $\operatorname{Hom}_G(F(M),F(N))$ is isomorphic to the subspace of $F(N \times M)$ consisting of those functions $K$ which are constant on orbits. The dimension of this subspace is the number of orbits, because it has a basis consisting of functions which take the value $1$ on a particular orbit and $0$ everywhere else.