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

Author

Listed:
  • 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)

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.

Suggested Citation

  • Anton Dzhamay & Yang Shi & Alexander Stokes & Ralph Willox, 2025. "What Is the Symmetry Group of a d-P II Discrete Painlevé Equation?," Mathematics, MDPI, vol. 13(7), pages 1-28, March.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:7:p:1123-:d:1623209
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