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Dominance regions for affine cluster algebras

Nathan Reading, Dylan Rupel, Salvatore Stella

TL;DR

This work characterizes dominance regions for cluster algebras of affine type, proving that these regions are line segments parallel to the imaginary ray when projected to the appropriate g-vector framework, and are points in finite-type cases. The authors develop a robust toolkit—mutation fans, imaginary walls, folding via stable automorphisms, and neighboring-seed analysis—to reduce and linearize dominance regions, and they extend these results from coefficient-free to coefficient-bearing and extended (tall) matrices. Central contributions include a detailed description of neighboring seeds, the role of the imaginary ray in affine geometry, and a cohesive affine-type theory that unifies acyclic and general seeds through folding and finite-type companions. The results illuminate integral dominance regions and connect to pointed bases and theta functions for affine cluster algebras, with potential implications for generalized minors and Cambrian-type combinatorics.

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).

Dominance regions for affine cluster algebras

TL;DR

This work characterizes dominance regions for cluster algebras of affine type, proving that these regions are line segments parallel to the imaginary ray when projected to the appropriate g-vector framework, and are points in finite-type cases. The authors develop a robust toolkit—mutation fans, imaginary walls, folding via stable automorphisms, and neighboring-seed analysis—to reduce and linearize dominance regions, and they extend these results from coefficient-free to coefficient-bearing and extended (tall) matrices. Central contributions include a detailed description of neighboring seeds, the role of the imaginary ray in affine geometry, and a cohesive affine-type theory that unifies acyclic and general seeds through folding and finite-type companions. The results illuminate integral dominance regions and connect to pointed bases and theta functions for affine cluster algebras, with potential implications for generalized minors and Cambrian-type combinatorics.

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).

Paper Structure

This paper contains 22 sections, 79 theorems, 47 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Dominance regions
  3. Definitions and background
  4. Folding the dominance region
  5. Linearizing the dominance region
  6. Dominance regions for extended exchange matrices
  7. Sufficient conditions for the dominance region to be a point
  8. Dominance regions in finite type
  9. Finite type, coefficient free
  10. Finite type, arbitrary coefficients
  11. Dominance regions in affine type
  12. Acyclic seeds of affine type
  13. General seeds of affine type
  14. Some tools
  15. Mutation-finite exchange matrices
  16. ...and 7 more sections

Key Result

Lemma 2.1

Suppose $\lambda'=\eta^{B^T}_{\boldsymbol{k}}(\lambda)$ and $B'=\mu_{\boldsymbol{k}}(B)$.

Figures (4)

  • Figure 1: A bipartite triangulation $T$ and the lamination $\Theta^{(\ell)}$
  • Figure 2: An illustration of the proof of the Type-A case
  • Figure 3: A bipartite tagged triangulation $T$ and lamination $\Theta^{(\ell)}$
  • Figure 4: Illustrations of the proof of Proposition \ref{['A and q-l']}

Theorems & Definitions (136)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 126 more