An unexpected property of $\mathbf{g}$-vectors for rank 3 mutation-cyclic quivers

Abstract

Let $Q$ be a rank 3 mutation-cyclic quiver. It is known that every $\mathbf{c}$-vector of $Q$ is a solution to a quadratic equation of the form $$\sum _{i=1}^3{x}_i^2+\sum _{1\le i<j\le 3}\pm q_{ij}{x}_i{x}_j=1,$$where $q_{ij}$ is the number of arrows between the vertices $i$ and $j$ in $Q$. A similar property holds for $\mathbf{c}$-vectors of any acyclic quiver. In this paper, we show that $\mathbf{g}$-vectors of $Q$ enjoy an unexpected property. More precisely, every $\mathbf{g}$-vector of $Q$ is a solution to a quadratic equation of the form $$\sum _{i=1}^3{x}_i^2+\sum _{1\le i<j\le 3}{p}_{ij}{x}_i{x}_j=1,$$where $p_{ij}$ is the number of arrows between the vertices $i$ and $j$ in another quiver $P$ obtained by mutating $Q$.

12 pages