The Lattice Structure of Connection Preserving Deformations for q-Painlev\'e Equations I

We wish to explore a link between the Lax integrability of the $q$-Painlev'e equations and the symmetries of the $q$-Painlev'e equations. We shall demonstrate that the connection preserving deformations that give rise to the $q$-Painlev'e equations may be thought of as elements of the groups of Schlesinger transformations of their associated linear problems. These groups admit a very natural lattice structure. Each Schlesinger transformation induces a B"acklund transformation of the $q$-Painlev'e equation. Each translational B"acklund transformation may be lifted to the level of the associated linear problem, effectively showing that each translational B"acklund transformation admits a Lax pair. We will demonstrate this framework for the $q$-Painlev'e equations up to and including $q$-${\mathrm{P}}_{\mathrm{V}\mathrm{I}}$.