My question is as follows: is the F-polynomial of an indecomposable quiver representation irreducible as a polynomial, i.e. can it have a nontrivial factor? Here the F-polynomial is the generating function of the Euler characteristics of quiver Grassmannians, that is,

F_M =\sum_{e_1,...e_n} \chi( Gr_{e_1,...e_n} M) x_1^e_1 ... x_n^e_n

It is known that the F-polynomial of the direct sum of $M_1$ and $M_2$ is the product of $F_{M_1}$ and $F_{M_2}$.

Question. Is the $F$-polynomial of an indecomposable quiver representation irreducible?

Here the $F$-polynomial is the generating function of the Euler characteristics of quiver Grassmannians, that is,

$$F_M =\sum_{e_1,...e_n} \chi( \mathrm{Gr}_{e_1,...e_n} M) x_1^{e_1} ... x_n^{e_n}$$

It is known that the $F$-polynomial of the direct sum of $M_1$ and $M_2$ is the product of $F_{M_1}$ and $F_{M_2}$.