Quantum Cluster Characters

TL;DR

This paper proves that quantum cluster characters establish a bijection between indecomposable rigid valued representations and quantum cluster variables, also showing that Grassmannians of subrepresentations have counting polynomials.

Contribution

It confirms the conjecture that quantum cluster characters provide a bijection to quantum cluster variables and demonstrates the polynomial countability of Grassmannians of subrepresentations.

Findings

Quantum cluster characters bijectively correspond to quantum cluster variables.

Grassmannians of subrepresentations have counting polynomials.

The approach generalizes Hubery's method to valued quivers.

Abstract

Let $\FF$ be a finite field and $(Q,\bfd)$ 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 $\cA_{|\FF|}(\tilde{B},\Lambda)$. As a corollary we find that, for any rigid valued representation $V$ of $Q$, all Grassmannians of subrepresentations $Gr_\bfe^V$ have counting polynomials.