For a Zamolodchikov periodic cluster algebra, the bipartite belt of bipartite mutations is periodic, and tropicalizing its T-system colors each vertex-mutation red (a Γ\Gamma-mutation) or blue (a Δ\Delta-mutation) according to which of the two sums attains the maximum in the max-plus exchange relation. Chin conjectured that for every admissible Dynkin biagram (Γ,Δ)(\Gamma,\Delta) with rr vertices and Coxeter numbers hΓ,hΔh_\Gamma,h_\Delta, one full period contains exactly hΓ⋅rh_\Gamma\cdot r red and hΔ⋅rh_\Delta\cdot r blue mutations — precisely the number of roots of the Γ\Gamma- and Δ\Delta-components. The conjecture is verified only numerically (SageMath, small ranks) and proved only in the single-Dynkin case (an arbitrary Dynkin diagram Λ\Lambda viewed as Λ⊗A1\Lambda\otimes A_1) under the ordering λi<0\lambda_i<0, where the half-period colored mutations moreover biject with the almost positive roots of Λ\Lambda. The problem is to prove or refute the conjecture for all admissible Dynkin biagrams and generic initial labelings.

Background

Cluster algebras, introduced by Fomin and Zelevinsky, are commutative rings built from an initial seed (x,B)(\mathbf{x},B), where B=(bij)B=(b_{ij}) is a skew-symmetrizable integer matrix (a B-matrix: some positive diagonal rescaling makes it skew-symmetric) and x=(x1,…,xn)\mathbf{x}=(x_1,\dots,x_n) is a free generating set of a field. A seed mutation μk\mu_k replaces xkx_k by xk′x_k' through an exchange relation and updates BB by a matrix mutation rule. A matrix BB is bipartite if its index set carries a coloring ϵ:[n]→{∘,∙}\epsilon:[n]\to\{\circ,\bullet\} such that bij=0b_{ij}=0 whenever i,ji,j have the same color; mutations at unconnected indices commute, so one may mutate all white vertices and all black vertices simultaneously, obtaining the bipartite mutations μ∘,μ∙\mu_\circ,\mu_\bullet and the bipartite belt μ∘μ∙μ∘μ∙⋯\mu_\circ\mu_\bullet\mu_\circ\mu_\bullet\cdots of Fomin and Zelevinsky. The matrix is recurrent if each bipartite mutation returns −B-B (that is, μ∘(B)=μ∙(B)=−B\mu_\circ(B)=\mu_\bullet(B)=-B, so μ∘μ∙(B)=B\mu_\circ\mu_\bullet(B)=B), and a bipartite recurrent matrix whose belt is periodic is called Zamolodchikov periodic — a property first observed by Zamolodchikov in the thermodynamic Bethe ansatz (a method for extracting finite-size spectra of integrable quantum field theories from their scattering data) for reflectionless ADE scattering theories, where the related Y-system functional relations are periodic with periods tied to Coxeter numbers (the Coxeter number of a Dynkin root system is the order of a Coxeter element of its reflection group, equivalently its number of roots divided by its rank). Along the belt the cluster variables satisfy the T-system Tk(t+1)Tk(t−1)=∏iTi(t)Γik+∏jTj(t)ΔjkT_k(t+1)T_k(t-1)=\prod_i T_i(t)^{\Gamma_{ik}}+\prod_j T_j(t)^{\Delta_{jk}}, a discrete dynamical system that is the cluster-algebraic form of generalized Y-systems.

The two papers at the origin of this problem organize the dynamics through Dynkin biagrams: a pair (Γ,Δ)(\Gamma,\Delta) of (not necessarily irreducible) n×nn\times n Coxeter adjacency matrices of Dynkin type whose sum B=Γ+ΔB=\Gamma+\Delta is bipartite and which share no nonzero entries. The biagram is admissible if the associated Cartan matrices 2I−Γ2I-\Gamma and 2I−Δ2I-\Delta commute (equivalently ΓΔ=ΔΓ\Gamma\Delta=\Delta\Gamma). Chin's classification theorem states that the Zamolodchikov periodic B-matrices are exactly those arising from admissible Dynkin biagrams, classified into 29 infinite families and 14 exceptional types beyond Stembridge's classification of commuting simply-laced Cartan matrices (6 infinite families and 11 exceptional types), all generated from ADE types by two operations preserving Zamolodchikov periodicity: folding (quotienting by a bicolored automorphism) and taking the transpose. In a connected admissible biagram all components of Γ\Gamma share one Coxeter number hΓh_\Gamma and all components of Δ\Delta share one Coxeter number hΔh_\Delta; the T-system is periodic with Tk(t+2N)=Tk(t)T_k(t+2N)=T_k(t) for N=hΓ+hΔN=h_\Gamma+h_\Delta, so one full period is 2(hΓ+hΔ)2(h_\Gamma+h_\Delta) time steps, in which each of the rr vertices is mutated hΓ+hΔh_\Gamma+h_\Delta times (the minimal period may be a proper divisor of hΓ+hΔh_\Gamma+h_\Delta; the conjecture below counts over the full 2(hΓ+hΔ)2(h_\Gamma+h_\Delta)-step window, which is always a period). A root system of rank rr with Coxeter number hh has exactly h⋅rh\cdot r roots, so the conjectured mutation counts below are root counts.

The conjecture concerns the tropicalization of the T-system. For an initial labeling λ∈Rn\lambda\in\mathbb{R}^n (with tkλ(0)=tkλ(1)=λkt^\lambda_k(0)=t^\lambda_k(1)=\lambda_k), the tropical T-system is the max-plus recursion

tkλ(t+1)+tkλ(t−1)=max⁡(∑iΓik tiλ(t),  ∑jΔjk tjλ(t)),t^\lambda_k(t+1)+t^\lambda_k(t-1)=\max\Big(\sum_i\Gamma_{ik}\,t^\lambda_i(t),\;\sum_j\Delta_{jk}\,t^\lambda_j(t)\Big),

applied at each time step tt only to the vertices kk of the color class mutated at that step (white vertices at even tt, black vertices at odd tt), the remaining vertices being carried forward unchanged. For a generic labeling λ\lambda — a dense open condition — the maximum is attained by exactly one of the two sums at every mutation, so each mutation in the period can be colored: red (a Γ\Gamma-mutation) if the Γ\Gamma-sum wins, blue (a Δ\Delta-mutation) if the Δ\Delta-sum wins. Numerical experiments led Chin to conjecture a precise link with root systems: in one full period the number of Γ\Gamma-mutations is hΓ⋅rh_\Gamma\cdot r, the total number of roots of the root systems of the Γ\Gamma-components, and likewise the number of Δ\Delta-mutations is hΔ⋅rh_\Delta\cdot r.

Two partial results frame the problem. First, the conjecture has been verified in SageMath on all non-ADE families of the classification for small ranks and on all infinite ADE families for n≤10n\le 10. Second, the half-periodicity theorem of the 2026 paper proves that for every Zamolodchikov periodic cluster algebra the form at the half-period t+hΓ+hΔt+h_\Gamma+h_\Delta is a permutation of order at most two of the cluster variables — closing, in general, a question conjectured by Kuniba, Nakanishi and Suzuki for Y-systems and proved by Inoue, Iyama, Keller, Kuniba and Nakanishi for tensor products of two simply-laced Dynkin diagrams — and its Corollary 5.2 verifies the count conjecture, together with a bijection, in the single-Dynkin case: an arbitrary Dynkin diagram Λ\Lambda of rank rr and Coxeter number hh, viewed as the tensor-product biagram Λ⊗A1\Lambda\otimes A_1 (with Γ=Λ\Gamma=\Lambda and Δ\Delta consisting of rr copies of A1A_1), under the global ordering λi<0\lambda_i<0 for all ii (the tropical shadow of xi≪1x_i\ll 1). There, one period contains exactly hrhr red and 2r2r blue mutations, and in one half period the blue mutations are in bijection with the negative simple roots of Λ\Lambda and the red mutations with its positive roots — together, the almost positive roots, which by Fomin and Zelevinsky's finite-type classification parametrize the cluster variables of any finite-type cluster algebra. Nothing is proved for any other family, for other orderings, or for generic labelings.

Problem Statement

Let (Γ,Δ)(\Gamma,\Delta) be an admissible Dynkin biagram with rr vertices and well-defined Coxeter numbers hΓ,hΔh_\Gamma,h_\Delta (in a connected biagram all Γ\Gamma-components share one Coxeter number hΓh_\Gamma and all Δ\Delta-components share one hΔh_\Delta; a disconnected biagram is handled component by component), and let tλt^\lambda be its tropical T-system with a generic initial labeling λ∈Rr\lambda\in\mathbb{R}^r (generic meaning the maximum in the tropical recursion is attained by exactly one of the two sums at every mutation of the period). Color each vertex-mutation in one full period — the 2(hΓ+hΔ)2(h_\Gamma+h_\Delta) time steps over which the tropical T-system returns to its initial state, during which each vertex is mutated hΓ+hΔh_\Gamma+h_\Delta times, the vertices of one color class being mutated at each step (white at even steps, black at odd steps) with the others carried forward unchanged — red if the Γ\Gamma-sum attains the maximum and blue if the Δ\Delta-sum does.

Prove or refute Chin's conjecture: for every admissible Dynkin biagram — equivalently, every Zamolodchikov periodic B-matrix in the classification (all 29 infinite families, 14 exceptional types, and Stembridge's simply-laced families of 6 infinite families and 11 exceptional types) — and every generic initial labeling, one full period contains exactly hΓrh_\Gamma r red mutations and hΔrh_\Delta r blue mutations; that is, the Γ\Gamma-mutations count the total number of roots of the root systems of the Γ\Gamma-components and the Δ\Delta-mutations count the total number of roots of the Δ\Delta-components. The restrictive ordering assumption of the known special case (Λ⊗A1\Lambda\otimes A_1 with λi<0\lambda_i<0) is removed: the claim concerns generic labelings across the full classified family list.

A complete answer proves or refutes the conjecture over the full classification without the ordering restriction; a refutation by an explicit finite counterexample (biagram plus generic labeling) is equally complete. Proofs for additional subfamilies beyond Λ⊗A1\Lambda\otimes A_1 (for example all tensor products, or all simply-laced biagrams) are progress but not completion.

References

  1. Ariana Chin. Half-periodicity of Zamolodchikov periodic cluster algebras. arXiv:2602.15140 (2026). doi:10.48550/arXiv.2602.15140. https://arxiv.org/abs/2602.15140
  2. Ariana Chin. Classification of Zamolodchikov periodic cluster algebras. arXiv:2510.18031 (2025). doi:10.48550/arXiv.2510.18031. https://arxiv.org/abs/2510.18031
  3. Ariana Chin. Classification of Zamolodchikov periodic cluster algebras. In Proceedings of the 38th Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2026, Seattle), Séminaire Lotharingien de Combinatoire, Article #106 (2026). https://sites.math.washington.edu/fpsac2026/public/abstracts/chin.pdf
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