Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 010, 25 pages      arXiv:2006.00627      https://doi.org/10.3842/SIGMA.2021.010
Contribution to the Special Issue on Cluster Algebras

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

Su Ji Hong
Department of Mathematics, University of Nebraska-Lincoln, USA

Received June 01, 2020, in final form January 17, 2021; Published online February 01, 2021

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 to $Q$ 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 Lee-Lee conjecture for acyclic quivers of type $A$, $D$, and $E_{6}$ and $E_7$.

Key words: real Schur roots; $c$-vectors; acyclic quivers; non-self-crossing curves.

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