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Cluster Modular Groups of Affine and Doubly Extended Cluster Algebras

Dani Kaufman, Zachary Greenberg

TL;DR

This work develops a uniform framework for affine and doubly extended cluster algebras by introducing the $T_{\n,\nw}$ quivers and the scalar invariant $\chi(T_{\n,\nw})$. It shows the affine cluster modular group is $\Gamma_\tau\rtimes H$ and the doubly extended group is generated by this plus a new element $\delta$, with an overarching folding mechanism to relate folded/non-simply-laced cases. The authors construct finite quotients of the cluster complex, yielding affine and doubly extended generalized associahedra, and derive facet and cluster counts that illuminate the combinatorial structure of these infinite-type clusters. They also provide a detailed treatment of BC-type variants, surface interpretations via annuli and punctured disks, and extensive counting formulas, together with conjectures on topological types of the resulting associahedra. This framework enables systematic analysis of infinite mutation-type cluster algebras and their geometric incarnations, with potential connections to mapping class groups and surface triangulations.

Abstract

We calculate the cluster modular groups of affine and doubly extended typecluster algebras in a uniform way by introducing a new family of quivers. We use this uniformdescription to construct a natural finite quotient of the cluster complex of each affine anddoubly extended cluster algebra. Using this construction, we introduce the notion of affineand doubly extended generalized associahedra, and count their facets.

Cluster Modular Groups of Affine and Doubly Extended Cluster Algebras

TL;DR

This work develops a uniform framework for affine and doubly extended cluster algebras by introducing the quivers and the scalar invariant . It shows the affine cluster modular group is and the doubly extended group is generated by this plus a new element , with an overarching folding mechanism to relate folded/non-simply-laced cases. The authors construct finite quotients of the cluster complex, yielding affine and doubly extended generalized associahedra, and derive facet and cluster counts that illuminate the combinatorial structure of these infinite-type clusters. They also provide a detailed treatment of BC-type variants, surface interpretations via annuli and punctured disks, and extensive counting formulas, together with conjectures on topological types of the resulting associahedra. This framework enables systematic analysis of infinite mutation-type cluster algebras and their geometric incarnations, with potential connections to mapping class groups and surface triangulations.

Abstract

We calculate the cluster modular groups of affine and doubly extended typecluster algebras in a uniform way by introducing a new family of quivers. We use this uniformdescription to construct a natural finite quotient of the cluster complex of each affine anddoubly extended cluster algebra. Using this construction, we introduce the notion of affineand doubly extended generalized associahedra, and count their facets.

Paper Structure

This paper contains 32 sections, 33 theorems, 71 equations, 35 figures.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Acknowledgements
  4. Preliminaries on Dynkin Diagrams Quivers and Mutations
  5. Dynkin diagrams
  6. Quivers
  7. Quivers from Dynkin diagrams
  8. Cluster algebras from quivers
  9. Folding quivers and cluster algebras
  10. C-vectors and reddening sequences
  11. The Cluster Modular Group
  12. The cluster complex
  13. Computing cluster modular groups
  14. Reddening elements
  15. Folding cluster modular groups
  16. ...and 17 more sections

Key Result

Theorem 1.1

Let $\mathbf{n},\mathbf{w}$ be $m$ dimensional vectors of positive integers. Let $\chi(T_{\mathbf{n},\mathbf{w}}) = \sum(w_i(n_i^{-1}-1))+2$. Then we have the following: Moreover almostThe twisted Dynkin diagrams that are Langlands dual to standard diagrams have "dual" $T_{\mathbf{n},\mathbf{w}}$ quivers. However their cluster structure is identical to their duals, so we mostly don't need to trea

Figures (35)

  • Figure 1: Correspondence between oriented and weighted Dynkin diagrams.
  • Figure 2: A simple quiver before and after mutation.
  • Figure 3: The quiver mutation graphs for ${A}^{(1)}_2$.
  • Figure 4: The quiver $T'_{\mathbf{n},\mathbf{w}}$.
  • Figure 5: The quiver $T_{\mathbf{n},\mathbf{w}}$.
  • ...and 30 more figures

Theorems & Definitions (131)

  • Theorem 1.1
  • Theorem 1.2
  • Conjecture 1.3
  • Theorem 1.4: \ref{['thm:AffineCountingFacets']}
  • Conjecture 1.5
  • Example 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 121 more