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Integrable deformations of cluster maps of type $D_{2N}$

Wookyung Kim

TL;DR

This work extends a two-parameter integrable deformation of the type $D_{4}$ cluster map to a family of deformed maps of type $D_{2N}$ through Laurentification and a local expansion construction on quivers. It demonstrates that the deformed maps admit Laurentifications to high-rank cluster algebras and that their tropical (max-plus) degree-growth is quadratic, yielding zero algebraic entropy and supporting Liouville integrability. By iterating a local expansion, the authors build quivers with $4N$ nodes and show mutation periodicity up to permutation, enabling explicit cluster maps whose dynamics are governed by quadratic growth in the degree vectors. The results unify deformations of Dynkin-type cluster maps with a scalable quiver-theoretic framework, and they suggest deep connections to elliptic curves and Weyl-group actions, opening avenues for further exploration of discrete integrable systems in this setting.

Abstract

In this paper, we extend one of the main results from our joint work with Hone and Mase, in which we studied a deformed type $D_{4}$ map, to the general case of the type $D_{2N}$ for $N\geq3$. This can be achieved through a ``local expansion" operation, introduced in our joint work with Grabowski and Hone. This operation involves inserting a specific subquiver into the quiver arising from the Laurentification of the deformed type $D_{4}$ map. This insertion yields a new quiver, obtained through the Laurentification of the deformed type $D_{6}$ map and thus enables systematic generalization to higher ranks $D_{2N}$. We also study the degree growth of deformed type $D_{2N}$ map via the tropical method and conjecture that, for each $N$, the deformed map is an integrable, as indicated by the algebraic entropy test, the criterion for detecting integrability in the discrete dynamical systems.

Integrable deformations of cluster maps of type $D_{2N}$

TL;DR

This work extends a two-parameter integrable deformation of the type cluster map to a family of deformed maps of type through Laurentification and a local expansion construction on quivers. It demonstrates that the deformed maps admit Laurentifications to high-rank cluster algebras and that their tropical (max-plus) degree-growth is quadratic, yielding zero algebraic entropy and supporting Liouville integrability. By iterating a local expansion, the authors build quivers with nodes and show mutation periodicity up to permutation, enabling explicit cluster maps whose dynamics are governed by quadratic growth in the degree vectors. The results unify deformations of Dynkin-type cluster maps with a scalable quiver-theoretic framework, and they suggest deep connections to elliptic curves and Weyl-group actions, opening avenues for further exploration of discrete integrable systems in this setting.

Abstract

In this paper, we extend one of the main results from our joint work with Hone and Mase, in which we studied a deformed type map, to the general case of the type for . This can be achieved through a ``local expansion" operation, introduced in our joint work with Grabowski and Hone. This operation involves inserting a specific subquiver into the quiver arising from the Laurentification of the deformed type map. This insertion yields a new quiver, obtained through the Laurentification of the deformed type map and thus enables systematic generalization to higher ranks . We also study the degree growth of deformed type map via the tropical method and conjecture that, for each , the deformed map is an integrable, as indicated by the algebraic entropy test, the criterion for detecting integrability in the discrete dynamical systems.

Paper Structure

This paper contains 12 sections, 9 theorems, 141 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Deformation integrable cluster map associated with type $D_{4}$
  4. Periodic cluster map associated with type $D_{4}$
  5. Deformed map of type $D_{4}$
  6. Deformed cluster map of type $D_{6}$
  7. Laurentification of deformed type $D_{6}$ map
  8. Tropicalization and degree growth for deformed $\mathrm{D}_{6}$ map
  9. Local expansion
  10. Type $D_{2N}$ deformed cluster map
  11. Degree growth of deformed type $D_{2N}$ map
  12. Conclusion and future work

Key Result

Theorem 2.1

Every cluster variable generated by the cluster mutations is in the Laurent polynomial ring in its initial cluster variables.

Figures (11)

  • Figure 1: Quiver associated with special Somos-7 recurrence \ref{['specialsomos7']}
  • Figure 2: The extended quiver $Q_{A_{2}}$ constructed through Laurentification of \ref{['typeA2map']}
  • Figure 3: Extension from $Q_{A_{2}}$ to $Q_{A_{4}}$. Green nodes are new vertices which are inserted in $Q_{A_{2}}$ in the form of local expansion (shown in the figure \ref{['localexpansionA2toA4']})
  • Figure 4: Local expansion of the subquiver in $Q_{A_{2}}$. The nodes $c = \qty{3,4}$, $d=\qty{1,7}$, $c'=\qty{4,5}$ and $d'=\qty{2,10}$
  • Figure 5: Quiver, $Q_{A_{2N}}$ associated to deformation of type $A_{2N}$ cluster map grab. The red vertices indicate the nodes that already existed in $Q_{A_{4}}$ while the green vertices represent those newly included through local expansion.
  • ...and 6 more figures

Theorems & Definitions (21)

  • Definition 2.1: Cluster algebra
  • Example 2.1
  • Theorem 2.1: Laurent phenomenon (see FZ2002 )
  • Definition 2.2: Mutation periodic
  • Definition 2.3: Cluster map
  • Theorem 2.2: Zamolodchikov periodicity fz2fz4fr
  • Definition 2.4
  • Theorem 2.3: Symplectic reduction
  • Remark 3.1
  • Theorem 4.1: Laurentification of the deformed map
  • ...and 11 more