What Is the Symmetry Group of a d-P II Discrete Painlevé Equation?

Anton Dzhamay (), Yang Shi, Alexander Stokes and Ralph Willox
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Anton Dzhamay: Beijing Institute of Mathematical Sciences and Applications (BIMSA), No. 544, Hefangkou Village, Huaibei Town, Huairou District, Beijing 101408, China
Yang Shi: College of Science and Engineering, Flinders University, Flinders at Tonsley, Tonsley 5042, Australia
Alexander Stokes: Waseda Institute for Advanced Study (WIAS), Waseda University, 1–21–1 Nishi Waseda, Shinjuku-ku, Tokyo 169-0051, Japan
Ralph Willox: Graduate School of Mathematical Sciences, The University of Tokyo, 3–8–1 Komaba, Meguro-ku, Tokyo 153-8914, Japan

Mathematics, 2025, vol. 13, issue 7, 1-28

Abstract: The symmetry group of a (discrete) Painlevé 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 by using a specific example of a d-P II equation, which corresponds to a half-translation on the root lattice dual to its surface-type root lattice but 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 then 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.

Keywords: discrete integrable system; discrete Painlevé equation; symmetry group (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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