What Is the Symmetry Group of a d-P<sub>II</sub> Discrete Painlevé Equation?
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 classificati...
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-03-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/7/1123 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1850212703954534400 |
|---|---|
| author | Anton Dzhamay Yang Shi Alexander Stokes Ralph Willox |
| author_facet | Anton Dzhamay Yang Shi Alexander Stokes Ralph Willox |
| author_sort | Anton Dzhamay |
| collection | DOAJ |
| description | 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<sub>II</sub> 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. |
| format | Article |
| id | doaj-art-7d5c4a36f9404f1d9510a4b5908e71dd |
| institution | OA Journals |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-7d5c4a36f9404f1d9510a4b5908e71dd2025-08-20T02:09:17ZengMDPI AGMathematics2227-73902025-03-01137112310.3390/math13071123What Is the Symmetry Group of a d-P<sub>II</sub> Discrete Painlevé Equation?Anton Dzhamay0Yang Shi1Alexander Stokes2Ralph Willox3Beijing Institute of Mathematical Sciences and Applications (BIMSA), No. 544, Hefangkou Village, Huaibei Town, Huairou District, Beijing 101408, ChinaCollege of Science and Engineering, Flinders University, Flinders at Tonsley, Tonsley 5042, AustraliaWaseda Institute for Advanced Study (WIAS), Waseda University, 1–21–1 Nishi Waseda, Shinjuku-ku, Tokyo 169-0051, JapanGraduate School of Mathematical Sciences, The University of Tokyo, 3–8–1 Komaba, Meguro-ku, Tokyo 153-8914, JapanThe 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<sub>II</sub> 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.https://www.mdpi.com/2227-7390/13/7/1123discrete integrable systemdiscrete Painlevé equationsymmetry group |
| spellingShingle | Anton Dzhamay Yang Shi Alexander Stokes Ralph Willox What Is the Symmetry Group of a d-P<sub>II</sub> Discrete Painlevé Equation? Mathematics discrete integrable system discrete Painlevé equation symmetry group |
| title | What Is the Symmetry Group of a d-P<sub>II</sub> Discrete Painlevé Equation? |
| title_full | What Is the Symmetry Group of a d-P<sub>II</sub> Discrete Painlevé Equation? |
| title_fullStr | What Is the Symmetry Group of a d-P<sub>II</sub> Discrete Painlevé Equation? |
| title_full_unstemmed | What Is the Symmetry Group of a d-P<sub>II</sub> Discrete Painlevé Equation? |
| title_short | What Is the Symmetry Group of a d-P<sub>II</sub> Discrete Painlevé Equation? |
| title_sort | what is the symmetry group of a d p sub ii sub discrete painleve equation |
| topic | discrete integrable system discrete Painlevé equation symmetry group |
| url | https://www.mdpi.com/2227-7390/13/7/1123 |
| work_keys_str_mv | AT antondzhamay whatisthesymmetrygroupofadpsubiisubdiscretepainleveequation AT yangshi whatisthesymmetrygroupofadpsubiisubdiscretepainleveequation AT alexanderstokes whatisthesymmetrygroupofadpsubiisubdiscretepainleveequation AT ralphwillox whatisthesymmetrygroupofadpsubiisubdiscretepainleveequation |