Question

Let $Q$ be an acylic quiver. Let $E$ and $F$ be finite dimensional representations, with $E$ indecomposable. Suppose that, for some positive integer $r$, the representation $F$ injects into $E^{\oplus r}$. Suppose also that, for every vertex $v$ of $Q$, we have $\dim F_v \leq \dim E_v$.

Does it follow that $F$ injects into $E$?

There are results like this in the work of Derksen, Schofield and Weyman, but I can't find this particular statement. Thanks!

Answer 1

So I think you're asking if there is some kind of "saturation theorem" for generic rank.

The following is a counterexample. Let $Q$ be the ${\rm D}_4$ quiver with vertices 1,2,3,4 (4 is the center) where the orientation is $1 \to 4$, $2 \to 4$ and $4 \to 3$. Let $E$ be the unique indecomposable (up to isomorphism) of dimension $(1,1,1,2)$ and take $F$ to be the unique representation of dimension $(0,0,0,2)$. Now the simple $S_4$ injects into $E$ as the kernel of the map $E_4 \to E_3$, but $F$ does not inject into $E$. However, it does inject into $E \oplus E$.