Maximally reducible monodromy of bivariate hypergeometric systems

TL;DR

This paper characterizes when the monodromy of certain bivariate hypergeometric systems is maximally reducible, providing a complete criterion for the splitting of solution spaces into one-dimensional invariant subspaces.

Contribution

It establishes a necessary and sufficient condition for maximal reducibility of monodromy in Horn systems with specific geometric configurations.

Findings

Monodromy is maximally reducible under specific geometric conditions.

Characterization applies to Horn systems with simplicial configurations and particular Ore-Sato polygons.

Provides a complete criterion for the reducibility of monodromy representations.

Abstract

We investigate branching of solutions to holonomic bivariate hypergeometric systems of Horn's type. Special attention is paid to the invariant subspace of Puiseux polynomial solutions. We mainly study Horn systems defined by simplicial configurations and Horn systems whose Ore-Sato polygons are either zonotopes or Minkowski sums of a triangle and segments proportional to its sides. We prove a necessary and sufficient condition for the monodromy representation to be maximally reducible, that is, for the space of holomorphic solutions to split into the direct sum of one-dimensional invariant subspaces.