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

SIGMA 16 (2020), 049, 11 pages      arXiv:1807.03359      https://doi.org/10.3842/SIGMA.2020.049
Contribution to the Special Issue on Cluster Algebras

Reddening Sequences for Banff Quivers and the Class $\mathcal{P}$

Eric Bucher a and John Machacek b
a)  Department of Mathematics, Xavier University, Cincinnati, Ohio 45207, USA
b)  Department of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P3, Canada

Received June 15, 2019, in final form May 23, 2020; Published online June 08, 2020

Abstract
We show that a reddening sequence exists for any quiver which is Banff. Our proof is combinatorial and relies on the triangular extension construction for quivers. The other facts needed are that the existence of a reddening sequence is mutation invariant and passes to induced subquivers. Banff quivers define locally acyclic cluster algebras which are known to coincide with their upper cluster algebras.The existence of reddening sequences for these quivers is consistent with a conjectural relationship between the existence of a reddening sequence and a cluster algebra's equality with its upper cluster algebra.Our result completes a verification of the conjecture for Banff quivers. We also prove that a certain subclass of quivers within the class $\mathcal{P}$ define locally acyclic cluster algebras.

Key words: cluster algebras; quiver mutation; reddening sequences.

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