Infinite-type cluster algebras have G-fans that are never complete, and rank 3 local behavior falls into six types that correlate with global fan shapes.
The existence of a maximal green sequence is not invariant under quiver mutation
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abstract
This note provides a quiver which does not admit a maximal green sequence, but which is mutation-equivalent to a quiver which does admit a maximal green sequence. The proof uses the `scattering diagrams' of Gross-Hacking-Keel-Kontsevich to show that a maximal green sequence for a quiver determines a maximal green sequence for any induced subquiver.
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Local and global patterns of rank 3 $G$-fans of totally-infinite type
Infinite-type cluster algebras have G-fans that are never complete, and rank 3 local behavior falls into six types that correlate with global fan shapes.