Linear relations for Laurent polynomials and lattice equations

A recurrence relation is said to have the Laurent property if all of its iterates are Laurent polynomials in the initial values with integer coefficients. Recurrences with this property appear in diverse areas of mathematics and physics, ranging from Lie theory and supersymmetric gauge theories to Teichmuller theory and dimer models. In many cases where such recurrences appear, there is a common structural thread running between these different areas, in the form of Fomin and Zelevinsky's theory of cluster algebras. Laurent phenomenon algebras, as defined by Lam and Pylyavskyy, are an extension of cluster algebras, and share with them the feature that all the generators of the algebra are Laurent polynomials in any initial set of generators (seed). Here we consider a family of nonlinear recurrences with the Laurent property, referred to as "Little Pi", which was derived by Alman et al. via a construction of periodic seeds in Laurent phenomenon algebras, and generalizes the Heideman-Hogan family of recurrences. Each member of the family is shown to be linearizable, in the sense that the iterates satisfy linear recurrence relations with constant coefficients. We derive the latter from linear relations with periodic coefficients, which were found recently by Kamiya et al. from travelling wave reductions of a linearizable lattice equation on a 6-point stencil. By making use of the periodic coefficients, we further show that the birational maps corresponding to the Little Pi family are maximally superintegrable. We also introduce another linearizable lattice equation on the same 6-point stencil, and present the corresponding linearization for its travelling wave reductions. Finally, for both of the 6-point lattice equations considered, we use the formalism of van der Kamp to construct a broad class of initial value problems with the Laurent property.