Séminaire Lotharingien de Combinatoire, 95B.106 (2026), 12 pp.
Ariana Chin
Classification of Zamolodchikov Periodic Cluster Algebras
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 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 natural 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 also helps classify all nonnegative W-cells for
products of two dihedral groups W
= I2(p) × I2(q).
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
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