Dominance regions for affine cluster algebras

TL;DR

This paper characterizes dominance regions in affine cluster algebras, revealing they are often line segments, and develops new tools to analyze neighboring seeds near the boundary of the g-vector fan.

Contribution

It explicitly describes dominance regions for affine cluster algebras and introduces a new method for analyzing neighboring seeds close to the g-vector fan boundary.

Findings

Dominance regions are often line segments in affine types.

Explicit descriptions of these regions are provided.

New tools for analyzing neighboring seeds are developed.

Abstract

We determine dominance regions associated to cluster algebras of affine type. In the most interesting cases, the dominance region is a line segment, which we describe explicitly. Motivations for this work include a project to determine all pointed bases for cluster algebras of affine type and a separate project to determine all theta functions in the affine case. The proofs draw on known results from the doubled Cambrian fan and almost-positive roots models, as well as a new tool that we develop: a detailed description of neighboring seeds of affine type (seeds that are, in some sense, as close as possible to the boundary of the g-vector fan).