Denseness of $g$-vector cones from weighted orbifolds

TL;DR

This paper investigates the structure of g-vector cones in cluster algebras derived from weighted orbifolds, revealing their union's closure and identifying special cases with half-space configurations.

Contribution

It characterizes the closure of the union of g-vector cones for weighted orbifolds, providing explicit descriptions and identifying exceptional cases.

Findings

Union of g-vector cones is dense in bR^n except for a special orbifold case.

In the special case, the union's closure is a half space defined by an explicit hyperplane.

The results extend understanding of cluster algebra combinatorics from orbifold surfaces.

Abstract

We study $g$-vector cones in a cluster algebra defined from a weighted orbifold of rank $n$ introduced by Felikson, Shapiro and Tumarkin. We determine the closure of the union of the $g$-vector cones. It is equal to $\mathbb{R}^n$ except for a weighted orbifold with empty boundary and exactly one puncture, in which case it is equal to the half space of a certain explicit hyperplane in $\mathbb{R}^n$.