A Lax Formulation of a Generalized q-Garnier System

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