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

Abstract

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 with 𝑄 are the dimension vectors of the indecomposable exceptional modules. It has been shown in [Nájera Chávez 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 the Lee-Lee conjecture for acyclic quivers of type 𝐴, 𝐷, and 𝐸₆ and 𝐸₇.