What is the symmetry group of a d-${\text{P}_{\mathrm{II}}}$ discrete Painlev\'e equation?

Abstract

The symmetry group of a (discrete) Painlev\'e equation provides crucial information on the properties of the equation. In this paper we argue against the commonly-held belief that the symmetry group of a given equation is solely determined by its surface type as given in the famous Sakai classification. We will dispel this misconception on a specific example of a d-${\text{P}_{\mathrm{II}}}$ equation which corresponds to a half-translation on the root lattice dual to its surface-type root lattice, but which becomes a genuine translation on a sub-lattice thereof that corresponds to its real symmetry group. The latter fact is shown in two different ways: first by a brute force calculation and second through the use of normalizer theory, which we believe to be an extremely useful tool for this purpose. We finish the paper with the analysis of a sub-case of our main example which arises in the study of gap probabilities for Freud unitary ensembles, and the symmetry group of which is even further restricted due to the appearance of a nodal curve on the surface on which the equation is regularized.