Dominance regions for affine cluster algebras

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 application that determines 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).