Finite orbits of monodromies of rank two Fuchsian systems

TL;DR

This paper classifies all finite monodromy orbits for five 2x2 matrices in Fuchsian systems, providing explicit proofs and methods that could extend to more matrices, advancing understanding of algebraic solutions of the Garnier system.

Contribution

It offers the first complete classification of finite monodromy orbits for five matrices and proposes a conjecture for higher numbers, introducing new methods for analysis.

Findings

Classified finite monodromy orbits for five matrices

Developed an induction method for larger matrices

Proposed a conjecture for six or more matrices

Abstract

We classified finite orbits of monodromies of the Fuchsian system for five $2\times 2$ matrices. The explicit proof of this result is given. We have proposed a conjecture for a similar classification for $6$ or more $2\times 2$ matrices. Cases in which all monodromy matrices have a common eigenvector are excluded from the consideration. To classify the finite monodromies of the Fuchsian system we combined two methods developed in this paper. The first is an induction method: using finite orbits of smaller number of monodromy matrices the method allows the construction of such orbits for bigger numbers of matrices. The second method is a formalism for representing the tuple of monodromy matrices in a way that is invariant under common conjugation way, this transforms the problem into a form that allows one to work with rational numbers only. The classification developed in this paper can be considered as the first step to a classification of algebraic solutions of the Garnier system.