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


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 $F\_{M\_2}$.



Irreducibility of the F-polynomial of an indecomposable quiver representation
No.


Consider the Kronecker quiver: two vertices with two arrows between them. Consider the representation with dimension vector $(3,3)$ given by the matrices
$$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \quad 
\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}.$$


The corresponding $F$-polynomial is computed in section 5 of [Caldero-Zelevinsky](https://web.archive.org/web/20220119142357/https://arxiv.org/abs/math/0604054), up to a monomial change of variables. (I won't try to get the monomial change of variables right.) It is
$$\begin{matrix} y^3 & & & \\ + 3 y^2 & + x y^2 & & \\ + 3y & + 4 xy & +x^2 y & \\ +1 & +3x & +3x^2 & +x^3 \end{matrix}$$


This factors as
$$\begin{pmatrix} y & \\ +1 & + x \end{pmatrix} \begin{pmatrix} y^2 & & \\ +2y & & \\ +1 & + 2x & x^2 \end{pmatrix}.$$


Numerical experimentation reveals an interesting pattern: Let $F(d)$ be the $F$-polynomial of the indecomposable representation with dimension vector $(d,d)$. Then $F(n)$ appears to
be prime if and only if $n+1$ is prime. Presumably, this should be an obvious consequence of the Chebyshev polynomial formulas in [Sherman-Zelevinsky](https://web.archive.org/web/20220125012723/https://arxiv.org/abs/math/0307082), but I don't see it right now.


I can think of various ways you could try to salvage this, but I'll leave that project to you.


