Zamolodchikov integrability via rings of invariants

Zamolodchikov periodicity is periodicity of certein recursions associated with box products $X\mathrm{\square }Y$ of two finite type Dynkin diagrams. We suggest an affine analog of Zamolodchikov periodicity, which we call Zamolodchikov integrability. We conjecture that it holds for products $X\mathrm{\square }Y$, where $X$ is a finite type Dynkin diagram and $Y$ is an extended Dynkin diagram. We prove this conjecture for the case of $A_m\mathrm{\square }{A}_{2n-1}^{(1)}$. The proof employs cluster structures in certain classical rings of invariants, previously studied by S. Fomin and the author.