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On the Realization of Real Schur Roots as Planar Curves

Su Ji Hong, University of Nebraska - Lincoln


In the intersection of representation theory and cluster algebra theory lies real Schur roots. Given an acyclic quiver Q and an algebraically closed field k, these roots are the dimension vectors of rigid indecomposable representations of the path algebra kQ. There are multiple ways to identify the real Schur roots among positive roots. It is known that the set of real Schur roots coincides with the set of positive c-vectors and the set of d-vectors of non-initial cluster variables. In an attempt to find a diagrammatic description for the real Schur roots, K.-H. Lee and K. Lee conjectured that the set of real Schur roots and the set of roots corresponding of non-self-crossing admissible curves coincide for acyclic quivers. We further this conjecture to show that for certain quivers, the set of associated roots of non-decreasing non-self-crossing admissible curves is equivalent to the set of real Schur roots.

Subject Area


Recommended Citation

Hong, Su Ji, "On the Realization of Real Schur Roots as Planar Curves" (2021). ETD collection for University of Nebraska - Lincoln. AAI28419695.