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arXiv · 2512.02218

Dominance regions for affine cluster algebras

Nathan Reading·Dylan Rupel·Salvatore Stella
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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).

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Nathan Reading, Dylan Rupel, Salvatore Stella. 2025-12-01. Dominance regions for affine cluster algebras. https://arxiv.org/abs/2512.02218

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