Acyclic cluster algebras with dense $g$-vector fans

TL;DR

This paper characterizes when acyclic cluster algebras have dense $g$-vector fans, showing they occur only in finite or affine types, and applies this to classify hereditary algebras with dense fans.

Contribution

It provides a complete classification of acyclic cluster algebras with dense $g$-vector fans and links this property to algebra type, advancing understanding of cluster algebra geometry.

Findings

Dense $g$-vector fans occur only in finite or affine types.

Acyclic cluster algebra has dense $g$-vector fan iff it is finite or affine.

Classification of hereditary algebras with dense $g$-vector fans.

Abstract

The $g$-vector fans play an important role in studying cluster algebras and silting theory. We survey cluster algebras with dense $g$-vector fans and show that a connected acyclic cluster algebra has a dense $g$-vector fan if and only if it is either finite type or affine type. As an application, we classify finite dimensional hereditary algebras with dense $g$-vector fans.