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

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2021
• Main Author: Hong, Su Ji
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2021
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/211178
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: -Vectors and Non-Self-Crossing Curves for Acyclic Quivers of Finite Type. Su Ji Hong. SIGMA 17 (2021), 010, 25 pages

Let be an acyclic quiver and k be an algebraically closed field. The indecomposable exceptional modules of the path algebra have been widely studied. The real Schur roots of the root system associated with 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 -vectors and the set of real Schur roots coincide. To give a diagrammatic description of -vectors, K-H. Lee and K. Lee conjectured that for acyclic quivers, the set of -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 , , and ₆ and ₇.