A continuous limit of the 𝑞-Garnier system (Recent developments in mathematics of integrable systems)

Abstract

In a recent work [5], the 𝑞-Garnier system was formulated in a framework of an extended affine Weyl group with the aid of cluster mutations. This fact allows us to understand that the 𝑞-Garnier system is a 𝑞-difference system of multi-directions governed by the theory of the affine root systems. It is known that a direction of the 𝑞-Garnier system reduces to the Hamiltonian system proposed in [1, 7, 10]. In this article we give the group of symmetries of the Hamiltonian system from the extended affine Weyl group by taking a continuous limit 𝑞 → 1.