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Half-periodicity of Zamolodchikov periodic cluster algebras

Ariana Chin

TL;DR

The paper proves half-periodicity for all Zamolodchikov periodic cluster algebras: at time $N = h_{\Gamma}+h_{\Delta}$ the associated $T$-system is carried to a permuted form $T_i(t+N) = T_{\sigma(i)}(t)$ by a permutation $\sigma$ of order at most two, with the permutation preserving or reversing bipartite color depending on the parity of $N$. The proof builds on showing the bipartite belt is a maximal green sequence, analyzing tropical $T$-systems, and then extending via folding and Langlands dual to all Zamolodchikov periodic $B$-matrices. It further connects half-periodicity to root-theoretic data, providing a colored mutation perspective that, in Dynkin cases under a global order, yields a bijection between almost positive roots and colored tropical mutations. These results generalize previous finite-type and tensor-product cases, offering a unified structural view of half-period dynamics in Zamolodchikov periodic cluster algebras and their $T$- and $Y$-system realizations.

Abstract

In 2007, Fomin and Zelevinsky introduced the bipartite belt, a sequence of bipartite mutations whose exchange relations form a discrete dynamical system. Periodicity of this system is known as Zamolodchikov periodicity. In our previous work we have classified all Zamolodchikov periodic cluster algebras, but behavior halfway through the period was still unknown. This so-called half-periodicity was conjectured by Kuniba--Nakanishi--Suzuki for $Y$-systems of finite type Cartan matrices, and was proved by Inoue--Iyama--Keller--Kuniba--Nakanishi for tensor products of two simply-laced Dynkin diagrams. In this paper, we prove that for any Zamolodchikov periodic cluster algebra, the form at the half-period is a permutation of the cluster variables of order at most two.

Half-periodicity of Zamolodchikov periodic cluster algebras

TL;DR

The paper proves half-periodicity for all Zamolodchikov periodic cluster algebras: at time the associated -system is carried to a permuted form by a permutation of order at most two, with the permutation preserving or reversing bipartite color depending on the parity of . The proof builds on showing the bipartite belt is a maximal green sequence, analyzing tropical -systems, and then extending via folding and Langlands dual to all Zamolodchikov periodic -matrices. It further connects half-periodicity to root-theoretic data, providing a colored mutation perspective that, in Dynkin cases under a global order, yields a bijection between almost positive roots and colored tropical mutations. These results generalize previous finite-type and tensor-product cases, offering a unified structural view of half-period dynamics in Zamolodchikov periodic cluster algebras and their - and -system realizations.

Abstract

In 2007, Fomin and Zelevinsky introduced the bipartite belt, a sequence of bipartite mutations whose exchange relations form a discrete dynamical system. Periodicity of this system is known as Zamolodchikov periodicity. In our previous work we have classified all Zamolodchikov periodic cluster algebras, but behavior halfway through the period was still unknown. This so-called half-periodicity was conjectured by Kuniba--Nakanishi--Suzuki for -systems of finite type Cartan matrices, and was proved by Inoue--Iyama--Keller--Kuniba--Nakanishi for tensor products of two simply-laced Dynkin diagrams. In this paper, we prove that for any Zamolodchikov periodic cluster algebra, the form at the half-period is a permutation of the cluster variables of order at most two.
Paper Structure (11 sections, 18 theorems, 24 equations, 2 figures)

This paper contains 11 sections, 18 theorems, 24 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Zamolodchikov periodicity
  3. The main theorem
  4. Preliminaries
  5. Introduction to cluster algebras with coefficients
  6. Maximal green sequences
  7. T-systems and Y-systems
  8. Tropical $T$-systems
  9. A maximal green sequence
  10. Half-periodicity of Zamolodchikov periodic cluster algebras
  11. Colored mutations

Key Result

Theorem 1.1

Let $(\Gamma, \Delta)$ be a Zamolodchikov periodic $n\times n$$B$-matrix with Coxeter numbers $h_{\Gamma}, h_{\Delta}$. Let $T(t)$ be the $T$-system associated to $(\Gamma, \Delta)$. Then for all $i\in [n]$, for some automorphism $\sigma\in S_n$ of $(\Gamma, \Delta)$ of order at most two. When $h_{\Gamma} + h_{\Delta}$ is even, this permutation is bipartite color preserving. Otherwise, it is colo

Figures (2)

  • Figure 1: The color preserving permutation of the cluster variables for $A_5\ast D_4$. On the left is the initial cluster at $t = 0$. On the right is the permuted cluster variables at the half-period $t = 10$.
  • Figure 2: The color reversing permutation of the cluster variables for $F_4\otimes A_2$. On the left is the initial cluster at $t = 0$. On the right is the permuted cluster variables at the half-period $t = 15$.

Theorems & Definitions (41)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: fomin
  • Remark 2.6
  • Theorem 2.7: Separation Formula fomin4
  • Proposition 2.8: kellerkeller2
  • Definition 2.9
  • ...and 31 more