Abstract
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 Teichmüller 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 six-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 six-point stencil, and present the corresponding linearization for its travelling wave reductions. Finally, for both of the six-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.
Highlights
There continues to be a great deal of interest in nonlinear recurrences of the form xn+mxn = P(xn+1, . . . , xm+n−1), (1)for a polynomial P, with the surprising property that all of the iterates are Laurent polynomials in the initial data with integer coefficients, that is to say xn ∈ Z[x±0 1, . . . , x±m−1 1]for all n
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
The Laurent property is an essential feature of the generators in cluster algebras, a novel class of commutative algebras introduced by Fomin and Zelevinsky [19], which are defined by recursive relations of the same form as (1) but with the restriction that P should be a binomial expression of a specific kind
Summary
(Note that, compared with [37], we have switched the order of the independent variables and introduced the parameter a.) By obtaining linear relations for the above lattice equation, they deduced linear recurrences with periodic coefficients for its (l, −k) travelling wave reduction (11) (cf proposition 3.5 and corollary 3.7 below) They proved the Laurent property for the lattice equation (17), in the sense that for the initial value problem defined by. I=1 where the arbitrary parameter b is an integration constant The latter family of recurrences was referred to as the ‘extreme polynomial’ in [2], where it was obtained from another set of period 1 seeds in LP algebras, and for b = 0 it was independently found in [52], where it was shown to be linearizable and have the Laurent property (see [35] for further details). We briefly review the construction of cluster algebras and LP algebras, and explain how particular recurrences of the form (1), having the Laurent property, appear in that context
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