# Irreducibility of the monodromy representation of Lauricella's $F_C$

## TL;DR

This paper proves that the monodromy representation of Lauricella's hypergeometric system is irreducible under specific parameter conditions and reducible otherwise, clarifying its algebraic structure.

## Contribution

It establishes the precise conditions for irreducibility of the monodromy representation of Lauricella's hypergeometric system.

## Key findings

- Monodromy representation is irreducible under 2^{m+1} parameter conditions.
- Representation becomes reducible if any condition is not satisfied.
- Provides a complete characterization of the monodromy's reducibility.

## Abstract

Let $E_C$ be the hypergeometric system of differential equations satisfied by Lauricella's hypergeometric series $F_C$ of $m$ variables. We show that the monodromy representation of $E_C$ is irreducible under our assumption consisting of $2^{m+1}$ conditions for parameters. We also show that the monodromy representation is reducible if one of them is not satisfied.