The monodromy representations of local systems associated with Lauricella's $F_D$

TL;DR

This paper explicitly describes the monodromy representations of local systems linked to Lauricella's hypergeometric system $F_D$, including reducible cases, and characterizes invariant subspaces via homology maps.

Contribution

It provides an effective description of monodromy representations for Lauricella's system, even when the system is reducible, and characterizes invariant subspaces through homology group maps.

Findings

Explicit monodromy representations for $F_D(a,b,c)$

Effective analysis in reducible cases

Characterization of invariant subspaces via homology maps

Abstract

We give the monodromy representations of local systems of twisted homology groups associated with Lauricella's system $F_D(a,b,c)$ of hypergeometric differential equations under mild conditions on parameters. Our representation is effective even in some cases when the system $F_D(a,b,c)$ is reducible. We characterize invariant subspaces under our monodromy representations by the kernel or image of a natural map from a finite twisted homology group to locally finite one.