Reddening Sequences for Banff Quivers and the Class
arXiv:1807.03359 · doi:10.3842/SIGMA.2020.049
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 define locally acyclic cluster algebras.
References in corpus (15)
- Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
- Total positivity, Grassmannians, and networks
- Locally acyclic cluster algebras
- A=U for Locally Acyclic Cluster Algebras
- Cohomology of cluster varieties. I. Locally acyclic case
- On cluster algebras from once punctured closed surfaces
- Donaldson-Thomas trasnsformations of moduli spaces of G-local systems
- Cluster algebra structures on Poisson nilpotent algebras
- Maximal Green Sequences for Cluster Algebras Associated to the n-Torus
- On Maximal Green Sequences For Type A Quivers
- Maximal Green Sequences for Cluster Algebras Associated to the Orientable Surfaces of Genus n with Arbitrary Punctures
- Upper cluster algebras and choice of ground ring
- Uniformly column sign-coherence and the existence of maximal green sequences
- Maximal green sequences of skew-symmetrizable 3x3 matrices
- On the relationship between green-to-red sequences, local-acyclicity, and upper cluster algebras