We follow Alexander Kirilov's book "Quiver Representations and Quiver Varieties", Section 8.3: McKay correspondence: Let $G$ be a nontrivial finite subgroup in $SU(2)$. Let $Q(G)$ be the finite graph with set of vertices $V := \operatorname{Irr}(G)$("isom classes of complex irreducible reps of $G$") and with the number of edges connecting vertices $i, j$ associated to irreps $\rho_i, \rho_j $ given by
$$ A_{ij}:= \dim \operatorname{Hom}_G(\rho_i, \rho_j \otimes \rho) $$
where $\rho$ the $2$-dimensional representation given by the inclusion $G \subset SU(2)$. Note that $\rho \cong \rho^*$ and that moreover that $\dim \operatorname{Hom}_G(\rho_i, \rho_j ) = \delta_{ij}$ by Maschke's Theorem.
On page 143 we are going to prove that $Q(R)$ is a connected graph. This is done in two essential steps I not fully understand.
Two Problems arise:
first, the autor states witout proof that the number of paths of length $n$ in $Q$ connecting $i$ with $j$ is equal to multiplicity of $\rho_i$ in $\rho_j \otimes \rho^{\otimes n}$. Why is this the case and how to see it? Say consider only the case $n=2$; ie number of paths of length $n=2$ in $Q$.
The claim is that it equals $\dim \operatorname{Hom}_G(\rho_i, \rho_j \otimes \rho^{\otimes 2})$ since this gives exactly the number of multiplicities $\rho_i$ occuring as direct factor in $\rho_j \otimes \rho^{\otimes 2}$.
Naive count of edges tells that the number of paths of length $2$ in $Q$ connecting $i$ with $j$ is $\sum_k A_{ik} \cdot A_{kj}$. Question is why it this the same as $\dim \operatorname{Hom}_G(\rho_i, \rho_j \otimes \rho^{\otimes 2})$?
The second step is the Lemma 8.14 & it's proof:
Lemma 8.14. Every irreducible representation of $G$ is a subrepresentation of $\rho^{\otimes n}$ for large enough $n$.
The proof argues as follows: consider the space $S^n \rho := \operatorname{Sym}^n(\mathbb{C}^2)$ of polynomials of degree $n$. Let $v \in \mathbb{C}^2$ be such that its stabilizer in $G$ is trivial; then the orbit $Gv$ is isomorphic to $G$, and the space $\mathbb{C}[Gv]$ of all functions on $Gv$ is the regular representation of $G$. On the other hand, since polynomials separate points, for large enough n the restriction map $S^n \rho \to \mathbb{C}[Gv]$ is surjective; thus, for such $n$, the space $S^n \rho$ (and hence the space $\rho^{\otimes n}$) contains every irreducible representation with nonzero multiplicity.
Problem 2: The last conclusion I not understand: Why the surectivity of $S^n \rho \to \mathbb{C}[Gv]$ implies that $S^n \rho$ contains every irrep as a direct factor? Is that the case for the regular representation $\mathbb{C}[G] \cong \mathbb{C}[Gv]$?