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Theory and Applications of the Elliptic Painlevé Equation

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Partition Functions and Automorphic Forms

Part of the book series: Moscow Lectures ((ML,volume 5))

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Abstract

This note is intended to provide an introduction to the theory of discrete Painlevé equations focusing mainly on the elliptic difference case. The elliptic Painlevé equation is the master case of the continuous/discrete Painlevé equations in two variables and has the largest affine Weyl group symmetry \(W(E_8^{(1)})\). Various “integrable” nature such as the symmetry of equation, bilinear form, Lax pair and special solutions are explained from a geometric point of view.

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Notes

  1. 1.

    From a geometric point of view [51], the third Painlevé equation P III is further classified into \(P_{\mathrm {III}_1}=P_{\mathrm {III}}^{D_6^{(1)}}\), \(P_{\mathrm {III}_2}=P_{\mathrm {III}}^{D_7^{(1)}}\) and \(P_{\mathrm {III}_3}=P_{\mathrm {III}}^{D_8^{(1)}}\).

  2. 2.

    Solving the Kovalevskaya’s top is much more difficult than the Euler/Lagrange cases because one need hyperelliptic functions.

  3. 3.

    Recently a method using the cluster algebra is also developed. See [4, 44] for example.

  4. 4.

    For the application to integrable systems, see the review [14] and references therein.

  5. 5.

    The conserved curve and the spectral curve can be identified for genus 1 case.

  6. 6.

    The representation (3.31) in variables (f i, g i) is algebraic but not \(\frak {S}_8\)-symmetric, while the representation (4.13) in variables (e i, h i, x, y) is \(\frak {S}_8\)-symmetric but not algebraic. There exist \(\frak {S}_8\)-symmetric and algebraic representations [1] which are, interestingly, related to the quadrirational Yang-Baxter maps.

  7. 7.

    The relation to the Eqs. (4.15) and (4.16) is given by similar way as the q-case Sect. 3.7.

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Acknowledgements

The author would like to thank his coworkers on various collaborations reviewed in this paper. He is also grateful to M. Bershtein, A. Grassi, A. Kels, V.P. Spiridonov, T. Suzuki, M. Yamazaki for interesting discussions. This work was supported by JSPS Kakenhi Grant (B) 26287018.

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  1. Department of Mathematics, Kobe Univ. Rokko, Kobe, Japan

    Yasuhiko Yamada

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  1. CNRS U.M.R. 8524, Université de Lille and IUF, Villeneuve d’Ascq Cedex, France

    Valery A. Gritsenko

  2. National Research University Higher School of Economics, Laboratory of Mirror Symmetry and Automorphic Forms, Moscow, Russia

    Vyacheslav P. Spiridonov

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Yamada, Y. (2020). Theory and Applications of the Elliptic Painlevé Equation. In: Gritsenko, V.A., Spiridonov, V.P. (eds) Partition Functions and Automorphic Forms. Moscow Lectures, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-030-42400-8_8

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