Symmetries of the Garnier system and of the associated polynomial Hamiltonian system
type: Item
instance of: scholarly article
title: Symmetries of the Garnier system and of the associated polynomial Hamiltonian system
zbMATH Open document ID: 0726.58025
author: Hironobu Kimura
published in: Proceedings of the Japan Academy. Series A
publication date: 1990-01-01T00:00:00Z
review text: For a completely integrable Pfaffian system \[ E(\theta):\;dx_ i=\sum^{n}_{j=1}G_{ij}(x,t,\theta)dt_ j,\quad G_{ij}\in {\mathbb{C}}(x,t),\quad i=1,...,m \] depending on parameters \(\theta \in {\mathbb{C}}^ N\), a symmetry for E(\(\theta\)) is a pair (S,l) of a birational transformation S: (x,t)\(\to (x',t')\) and an affine transformation l: \({\mathbb{C}}^ N\to {\mathbb{C}}^ N\) such that \((S^{- 1})^*E(\theta)=E(l(\theta))\). In this paper, a completely integrable Hamiltonian system \({\mathcal G}_ n\), with several time variables, and its associated Hamiltonian system \({\mathcal H}_ n\) are considered. The system \({\mathcal G}_ n\) is obtained by the monodromy preserving deformation of the second order Fuchsian equation on \({\mathbb{P}}^ 1\) and \({\mathcal H}_ n\) is derived from \({\mathcal G}_ n\) by a rational symplectic transformation. For the systems \({\mathcal G}_ n\) and \({\mathcal H}_ n\), a group of symmetries isomorphic to the Weyl group of type \(B_{n+3}\) is constructed.
reviewed by: Hironobu Kimura
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 58A17
Mathematics Subject Classification ID: 35Q58
Mathematics Subject Classification ID: 37J99
zbMATH DE Number: 4199084
zbMATH Keywords: completely integrable Hamiltonian system
zbMATH Keywords: monodromy preserving deformation
zbMATH Keywords: group of symmetries
zbMATH Keywords: Weyl group
MaRDI profile type: Publication
full work available at URL: pjaa.66.176
OpenAlex ID: W2050034786
cites work: On the polynomial Hamiltonian structure of the Garnier systems
cites work: ON PARTICULAR SOLUTIONS OF THE GARNIER SYSTEMS AND THE HYPERGEOMETRIC FUNCTIONS OF SEVERAL VARIABLES
cites work: Q3862133
cites work: Q3767632
cites work: Studies of the Painlevé equations. I: Sixth Painlevé equation \(P_{VI}\)
DOI: 10.3792/PJAA.66.176
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description: scientific article; zbMATH DE number 4199084