$C$-Vectors and Non-Self-Crossing Curves for Acyclic Quivers of Finite Type

Let $Q$ be an acyclic quiver and $k$ be an algebraically closed field. The indecomposable exceptional modules of the path algebra $kQ$ have been widely studied. The real Schur roots of the root system associated to $Q$ are the dimension vectors of the indecomposable exceptional modules. It has been shown in [N\'ajera Ch\'avez A., Int. Math. Res. Not. 2015 (2015), 1590-1600] that for acyclic quivers, the set of positive $c$-vectors and the set of real Schur roots coincide. To give a diagrammatic description of $c$-vectors, K-H. Lee and K. Lee conjectured that for acyclic quivers, the set of $c$-vectors and the set of roots corresponding to non-self-crossing admissible curves are equivalent as sets [Exp. Math., to appear, arXiv:1703.09113]. In [Adv. Math. 340 (2018), 855-882], A. Felikson and P. Tumarkin proved this conjecture for 2-complete quivers. In this paper, we prove a revised version of Lee-Lee conjecture for acyclic quivers of type $A$, $D$, and $E_{6}$ and $E_7$.