Quantum Cluster Characters

Let $\text{FF}$ be a finite field and $(Q,\text{bfd})$ an acyclic valued quiver with associated exchange matrix $\tilde{B}$. We follow Hubery’s approach \cite{hub1} to prove our main conjecture of \cite{rupel}: the quantum cluster character gives a bijection from the isoclasses of indecomposable rigid valued representations of $Q$ to the set of non-initial quantum cluster variables for the quantum cluster algebra ${\text{cA}}_{\mathrm{\mid }\text{FF}\mathrm{\mid }}(\tilde{B},\mathrm{\Lambda })$. As a corollary we find that, for any rigid valued representation $V$ of $Q$, all Grassmannians of subrepresentations $G{r}_{\text{bfe}}^V$ have counting polynomials.