Classification of Fuchsian systems and their connection problem

TL;DR

This paper reviews the Deligne-Simpson problem and demonstrates how middle convolutions can transform Fuchsian systems into fundamental systems with finite spectral types, providing explicit connection formulas for solutions.

Contribution

It introduces a method to transform Fuchsian systems into fundamental systems with finite spectral types using middle convolutions and provides explicit connection formulas.

Findings

Middle convolutions transform Fuchsian systems into fundamental systems.

Spectral types are confined to a finite set after transformation.

Explicit connection formulas for solutions are derived.

Abstract

We review the Deligne-Simpson problem, a combinatorial structure of middle convolutions and their relation to a Kac-Moody root system discoverd by Crawley-Boevey. We show with examples that middle convolutions transform the Fuchsian systems with a fixed number of accessory parameters into fundamental systems whose spectral type is in a finite set and we give an explicit connection formula for solutions of Fuchsian differential equations without moduli.