a better handle on the projective-free representations of $V_4$ by studying the representations of the so called Kronecker Quiver, which is a directed graph $Q$ on two vertices as shown below. $$\xymatrix{
 u_1 \bullet \ar@/^10pt/[rr]^f   \ar@/_10pt/[rr]_g &  & \bullet u_2
}$$ A representations of the above quiver is just a pair of finite dimensional $k$-vector spaces $V$ and $W$ and a pair of $k$-linear maps $\psi_1$ and $\psi_2$ from $V$ to $W$. Such a representation will be denoted by the four tuple $[V, W ; \psi_1, \psi_2]$, and given two such representations, the notion of direct sum, and morphisms between them are defined in the obvious way. Thus it makes sense to talk about the isomorphism class of an indecomposable representation of $Q$. Let us call a representation of $Q$ special if the following conditions hold: $$\begin{aligned}
\operatorname{Ker}(\psi_1) \;  \bigcap \; \operatorname{Ker}(\psi_2)  & = & 0 \\
{\text{Image}}(\psi_1) + {\text{Image}}(\psi_2) & = & V_2.\end{aligned}$$

\[prop1\] [@hell-rei] There is a one-one correspondence between the isomorphism classes of (indecomposable) projective-free representations of $V_4$ and those of the special (indecomposable) representations of the Kronecker quiver. Under this correspondence, an (indecomposable) projective-free representation $M$ of $G$ corresponds to the (indecomposable) representation of $Q$ that 