Hamiltonian reductions in Matrix Painlev\'e systems

TL;DR

This paper demonstrates how Hamiltonian reductions relate $G$-invariant Calogero--Painlevé particles and matrix Painlevé systems, revealing connections via folding transformations and symmetry reductions.

Contribution

It establishes a link between $G$-invariant Calogero--Painlevé systems and reduced matrix Painlevé systems through Hamiltonian reduction techniques.

Findings

Equivalence of $G$-invariant Calogero--Painlevé dynamics to $n$-particle systems.

Reduction of matrix Painlevé systems to $n imes n$ systems.

Use of folding transformations to classify reductions.

Abstract

For certain finite groups $G$ of B\"acklund transformations we show that the dynamics of $G$-invariant configurations of $n|G|$ Calogero--Painlev\'e particles is equivalent to certain $n$-particle Calogero--Painlev\'e system. We also show that the reduction of dynamics on $G$-invariant subset of $n|G|\times n|G|$ matrix Painlev\'e system is equivalent to certain $n\times n$ matrix Painlev\'e system. The groups $G$ correspond to folding transformations of Painlev\'e equations. The proofs are based on the Hamiltonian reductions.