Finite type and completeness of $g$-fans

TL;DR

This paper establishes a precise criterion linking the finiteness of skew-symmetrizable matrices in cluster algebras to the completeness of their associated $g$-fans, providing a geometric characterization of finite type.

Contribution

It proves that a skew-symmetrizable matrix is of finite type if and only if its $g$-fan is complete, connecting algebraic finiteness to geometric completeness.

Findings

Finite type matrices correspond to complete $g$-fans.

Completeness of $g$-fans is equivalent to their support containing all lattice points.

Provides a geometric criterion for classifying finite type cluster algebras.

Abstract

We study the $g$-fan associated with a skew-symmetrizable matrix in the sense of cluster algebras. We show that a skew-symmetrizable matrix is of finite type if and only if its $g$-fan is complete; equivalently (as we show), its support contains all lattice points.