Generalized $q$-Painlevé VI systems of type $(A_{2n+1}+A_1+A_1)^{(1)}$ arising from cluster algebra

Abstract

In this article we formulate a group of birational transformations which is isomorphic to an extended affine Weyl group of type (A 2n+1 + A 1 + A 1 ) (1) with the aid of mutations and permutations to a mutation-periodic quiver on a torus. This group provides four types of generalizations of Jimbo-Sakai's q-Painlevé VI equation as translations of the extended affine Weyl group. Then the known three systems are obtained again; the q-Garnier system, a similarity reduction of the lattice q-UC hierarchy and a similarity reduction of the q-Drinfeld-Sokolov hierarchy.

Discrete Hirota reductions associated with the lattice KdV equation

“…The T-systems ( 4) and ( 5) with constant coefficients can be seen as a particular case of ( 21) and (22), where β m = β ′ m = b for any m. So it follows that v m defined by ( 20) satisfies (19) whenever τ m satisfies one of these discrete Hirota reductions with constant coefficients, but the converse statement is not true.…”

“…It was shown in [14] that any bilinear difference equation of the form (2), or a suitable generalization with periodic coefficients, admits a Lax pair derived from the Lax representation of the discrete Hirota equation. In the cases of ( 21) and (22) this construction involves N × N and min(N, 2M) × min(N, 2M) Lax matrices respectively. However, in these cases there is also a 2 × 2 Lax representation derived from the Lax representation of the lattice KdV equation.…”

“…Now, for coprime N > M with N + M odd, we consider the second U-system (32), with periodic coefficients β ′ m+N = β ′ m , that corresponds to the discrete Hirota reduction (22). The map φ2 :…”

“…These kinds of recurrences inherit a Lax representation from the Lax representation of the discrete Hirota equation [14]. Their non-autonomous versions are associated with q-Painlevé equations and their higher order analogues [12,21,22], and they appear in the context of supersymmetric gauge theories and dimer models [1,2,3,9]. Furthermore, they are particular examples of cluster maps, which arise from cluster mutations of periodic quivers [7] and, as a consequence they exhibit the Laurent phenomenon, i.e.…”

Cluster realizations of Weyl groups and higher Teichmüller theory

“…An element of the cluster modular group which preserves the product of all X-variables gives rise to a discrete flow, and some interesting discrete integrable systems are realized in this way. In particular in [BGM17,OS18], discrete q-Painlevé equations are realized by Weyl groups which appear as the symmetries of the equations. For example, their quiver for the equation Ã3 (labeled by Sakai's classification) is identical to our quiver Q 2 ( Ã3 ) with the cyclically oriented Coxeter quiver, while the symmetry group is W ( D5 ).…”