Hankel Determinant Solutions to Several Discrete Integrable Systems and the Laurent Property

Abstract

Abstract. Many discrete integrable systems exhibit the Laurent phenomenon. In this paper, we investigate three integrable systems: the Somos-4 recurrence, the Somos-5 recurrence and a system related to so-called A 1 Q-system, whose general solutions are derived in terms of Hankel determinant. As a result, we directly confirm that they satisfy the Laurent property. Additionally, it is shown that the Somos-5 recurrence can be viewed as a specified Bäcklund transformation of the Somos-4 recurrence. The related topics about Somos polynomials are also studied.Key words. Hankel determinant solution, Laurent property, discrete integrable systems AMS subject classifications. 11B83, 37J35, 15B05, 11Y651. Introduction. Laurent phenomenon is a crucial property behind integrality shared by a class of combinatorial models while integrability is a key feature for a class of what we call integrable systems. An interesting observation is that many discrete integrable systems exhibit the Laurent phenomenon, and many mappings with the Laurent property are proved to be integrable. In the recent decade, a lot of related work were done via the study of a class of commutative algebras, called cluster algebras [19,21], for its success in proving the Laurent phenomenon. For instance, see [11,12,17,20,22,23,24,25,26,31,34]. More recently, a generalization of cluster algebras called Laurent phenomenon algebras [39] was also introduced in order to study the Laurent phenomenon [1,36].One question is whether we can prove the Laurent property (or a stronger property) from other views. In this paper, we shall give one possible choice-the view of explicit determinant solution. That is, the Laurent property of some discrete integrable systems can be proved by their determinant solutions. Of course, how to obtain the desired determinant formulae is another challenging problem. This approach succeeds for the following three integrable systems:The first two systems are the Somos-4 recurrence and the Somos-5 recurrence defined by

Some integrable maps and their Hirota bilinear forms

“…and conversely, the equation (5) for τ n , with the 2-periodic coefficient γ n , has a first integral β such that u n given by (4) satisfies (2). Moreover, if u n is given by (4), then when N is even (2) has a first integral K such that τ n satisfies…”

“…Moreover, if u n is given by (4), then when N is even (2) has a first integral K such that τ n satisfies…”

“…The complete proof of theorem 1.1, for arbitrary N, is reserved until section 3, where we begin by deriving (5) via a modified version of (2) (see equation (22) below), before treating the rest of the result and its detailed consequences for even/odd N separately. The interpretation in terms of Liouville integrability is naturally achieved by considering a lift of (2) to dimension N + 1, obtained by eliminating the parameter β: this yields equation (19) in section 3. A schematic picture of the connections between the main equations involved, valid for arbitrary N, is provided by the following diagram:…”

“…To see this method in practice, consider (2) for N = 5 with α = 3, β = −10, which gives the recurrence u n+5 = 10 − (u n + u n+1 + u n+2 + u n+3 + u n+4 ) + 3 u n+1 u n+2 u n+3 u n+4 , defined over Q, and choose the five rational initial data …”

“…Proposition 2.3. Given the substitution (4), u n is a solution of (2) whenever the tau function τ n satisfies the multilinear relation …”