Classification of Zamolodchikov periodic cluster algebras

TL;DR

This paper classifies all pairs of finite-type Cartan matrices that exhibit Zamolodchikov periodicity in cluster algebras, extending previous results and connecting to Kazhdan--Lusztig theory.

Contribution

It provides a complete classification of Zamolodchikov periodic cluster algebras via pairs of Cartan matrices, including new infinite families and exceptional types.

Findings

Classified all pairs of finite-type Cartan matrices with Zamolodchikov periodicity.

Identified that all such pairs derive from simply-laced types through folding and transpose operations.

Connected the classification to nonnegative W-cells in dihedral group products.

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, $W = I_2(p)\times I_2(q)$.