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Let $Q$ be the quiver with one vertex and one loop $\ell$, subject to the relation $\ell^2=0$. For this quiver, there are only two indecomposable representations (up to isomorphism): $$\xymatrix{ V_1 = & {\mathbb{C}}\ar@(ul,ur)^{0} & & \textrm{and} & & V_2 = & {\mathbb{C}}^{2} \ar@(ul,ur)^{\footnotesize  \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}}
  }$$ and only one almost-split sequence: $$0\to V_1\to V_2\to V_1 \to 0.$$ The $F$-polynomials are easily seen to be $F_{V_1}(y)=1+y$ and $F_{V_2}(y)=1+y+y^2$, and one can check that they satisfy Theorem \[theo::F-polynomials\].

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Let $Q$ and $V$ be as in Example \[exam::Grassmannians\] (\[exam::grassKron\]). Then $F_V(y_1, y_2)=1+2y_2+y_2^2+y_1y_2+2y_1y_2^2+y_1^2y_2^2$.

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We list a few more examples and properties of $F$-polynomials.

1.  If $V$ and $W$ are isomorphic, then $F_V=F_W$. The converse is false: consider the Kronecker quiver $$\xymatrix{ 1\ar@<.3ex>[r]^a \ar@<-.3ex>[r]_b  & 2.}$$ Then the representations $$\xymatrix{ V_1={\mathbb{C}}\ar@<.3ex>[r]^{\phantom{xx}0} \ar@<-.3ex>[r]_{\phantom{xx}1}  & {\mathbb{C}}& \textrm{and} & V_2={\mathbb{C}}\ar@<.3ex>[r]^{\phantom{xx}1} \ar@<-.3ex>[r]_{\phantom{xx}0} & {\mathbb{C}}}$$ are not isomorphic, but their $F$-polynomials are both equal to $1+y_2+y_1y_2$.

2.  If $F_V$ is an irreducible polynomial, then $V$ is indecomposable. The converse is false: consider the quiver $$\xymatrix{ 1\ar@<.3ex>[r]^a   & 2\ar@<.3ex>[l]^b.}$$ Then the representation $$\xymatrix{ {\mathbb{C}}^2\ar@<.3ex>[r]^{\footnotesize\begin{pmatrix} 1 & 0 \end{pmatrix}}   & {\mathbb{C}}\ar@<.3ex>[l]^{\footnotesize\begin{pmatrix} 0 \\ 1 \end{pmatrix}}.}$$ is indecomposable, but its $F$-polynomial is 