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Classification of Zamolodchikov periodic cluster algebras

Ariana Chin

TL;DR

The paper establishes a precise correspondence between Zamolodchikov periodicity in cluster algebras and pairs of commuting finite-type Cartan matrices, extending the known ADE framework. By introducing Dynkin biagrams and detailing folding, transposition, and binding operations, it achieves a complete classification into 29 infinite families and 14 exceptional types (beyond Stembridge) and shows these arise from simply-laced types via folding and transpose. A key bridge is built between periodicity, tropical T-systems, and strictly subadditive/fixed-point labelings, proving equivalences that ground the periodic behavior in Cartan data. The work further connects to Kazhdan-Lusztig theory through $W$-graphs and discusses implications for nonnegative $W$-cells in products of dihedral groups, while proposing conjectures on tropical mutations and root-system interpretations that point to rich future directions.

Abstract

Zamolodchikov periodicity is a property of certain discrete dynamical systems and was one of the primary motivations for the creation of cluster algebras. It was first observed by Zamolodchikov in his study of thermodynamic Bethe ansatz, initially for simply-laced Dynkin diagrams. It was proved by Keller to hold for tensor products of two Dynkin diagrams, and further shown by Galashin and Pylyavskyy to hold for pairs of commuting simply-laced Cartan matrices of finite type, which Stembridge classified in his study of admissible $W$-cells. We prove that the Zamolodchikov periodic cluster algebras are in bijection with pairs of commuting (not necessarily reduced or simply-laced) Cartan matrices of finite type. We fully classify all such pairs into 29 infinite families and 14 exceptional types in addition to the 6 infinite families and 11 exceptional types in Stembridge's classification, and show that all of these families can be derived from simply-laced types through two operations preserving Zamolodchikov periodicity, folding and taking transpose. Our work holds connections to Kazhdan--Lusztig theory, and our main theorem helps classify all nonnegative $W$-cells for products of two dihedral groups, $W = I_2(p)\times I_2(q)$.

Classification of Zamolodchikov periodic cluster algebras

TL;DR

The paper establishes a precise correspondence between Zamolodchikov periodicity in cluster algebras and pairs of commuting finite-type Cartan matrices, extending the known ADE framework. By introducing Dynkin biagrams and detailing folding, transposition, and binding operations, it achieves a complete classification into 29 infinite families and 14 exceptional types (beyond Stembridge) and shows these arise from simply-laced types via folding and transpose. A key bridge is built between periodicity, tropical T-systems, and strictly subadditive/fixed-point labelings, proving equivalences that ground the periodic behavior in Cartan data. The work further connects to Kazhdan-Lusztig theory through -graphs and discusses implications for nonnegative -cells in products of dihedral groups, while proposing conjectures on tropical mutations and root-system interpretations that point to rich future directions.

Abstract

Zamolodchikov periodicity is a property of certain discrete dynamical systems and was one of the primary motivations for the creation of cluster algebras. It was first observed by Zamolodchikov in his study of thermodynamic Bethe ansatz, initially for simply-laced Dynkin diagrams. It was proved by Keller to hold for tensor products of two Dynkin diagrams, and further shown by Galashin and Pylyavskyy to hold for pairs of commuting simply-laced Cartan matrices of finite type, which Stembridge classified in his study of admissible -cells. We prove that the Zamolodchikov periodic cluster algebras are in bijection with pairs of commuting (not necessarily reduced or simply-laced) Cartan matrices of finite type. We fully classify all such pairs into 29 infinite families and 14 exceptional types in addition to the 6 infinite families and 11 exceptional types in Stembridge's classification, and show that all of these families can be derived from simply-laced types through two operations preserving Zamolodchikov periodicity, folding and taking transpose. Our work holds connections to Kazhdan--Lusztig theory, and our main theorem helps classify all nonnegative -cells for products of two dihedral groups, .
Paper Structure (25 sections, 34 theorems, 99 equations, 23 figures, 4 tables)

This paper contains 25 sections, 34 theorems, 99 equations, 23 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Zamolodchikov periodicity
  3. The main theorem
  4. Preliminaries
  5. Cluster algebras
  6. $T$-systems and Zamolodchikov periodicity
  7. Tropical $T$-systems
  8. ADE bigraphs
  9. Dynkin biagrams
  10. Classification of commuting Cartan matrices as admissible Dynkin biagrams
  11. Bindings and folding
  12. Operations preserving admissibility
  13. Classification of admissible double bindings
  14. Proof of the classification
  15. A key observation
  16. ...and 10 more sections

Key Result

Theorem 1.1

The Zamolodchikov periodic $B$-matrices are in natural bijection with pairs of commuting Cartan matrices $(\Gamma, \Delta)$. The period divides $h_{\Gamma} + h_{\Delta}$, where $h_{\Gamma}, h_{\Delta}$ are the Coxeter numbers of $\Gamma, \Delta$.

Figures (23)

  • Figure 1: A nonadmissible ADE bigraph (left) and an admissible ADE bigraph (right). The vertices $v_1$ and $v_2$ form a nonadmissible pair.
  • Figure 2: The tensor product $D_4\otimes A_4$.
  • Figure 3: The twist $D_4\times D_4$.
  • Figure 4: The binding $E_6\ast E_6$.
  • Figure 5: The binding $A_7\ast D_5$.
  • ...and 18 more figures

Theorems & Definitions (88)

  • Theorem 1.1
  • Remark 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Definition 2.5
  • Example 2.6
  • Remark 2.7
  • Remark 2.8
  • ...and 78 more