A Lax formulation of a generalized $q$-Garnier system

TL;DR

This paper introduces a Lax formulation for a generalized $q$-Garnier system using a matrix-based $q$-difference equation framework, extending the understanding of integrable systems related to affine Weyl groups.

Contribution

It presents a novel Lax formulation of a generalized $q$-Garnier system based on a matrix $q$-difference equation approach, connecting cluster mutations and affine Weyl group symmetries.

Findings

Derived a Lax form for the generalized $q$-Garnier system.

Connected cluster mutation framework with $q$-difference equations.

Extended the class of integrable systems related to affine Weyl groups.

Abstract

Recently, a birational representation of an extended affine Weyl group of type $A_{mn-1}^{(1)}\times A_{m-1}^{(1)}\times A_{m-1}^{(1)}$ was proposed with the aid of a cluster mutation. In this article we formulate this representation in a framework of a system of $q$-difference equations with $mn\times mn$ matrices. This formulation is called a Lax form and is used to derive a generalization of the $q$-Garnier system.