Complex reflection groups as differential Galois groups

TL;DR

This paper introduces a method to construct integrable linear differential systems with a specified complex reflection group as their differential Galois group, providing explicit examples for many low-rank groups.

Contribution

It presents a new recipe for generating integrable systems with prescribed complex reflection groups as their differential Galois groups, expanding the understanding of their differential algebraic properties.

Findings

Explicit systems constructed for many low-rank groups

Demonstrates the realization of complex reflection groups as Galois groups

Provides a systematic approach for associating differential systems to reflection groups

Abstract

Complex reflection groups comprise a generalization of Weyl groups of semisimple Lie algebras, and even more generally of finite Coxeter groups. They have been heavily studied since their introduction and complete classification in the 1950s by Shephard and Todd, due to their many applications to combinatorics, representation theory, knot theory, and mathematical physics, to name a few examples. For each given complex reflection group G, we explain a new recipe for producing an integrable system of linear differential equations whose differential Galois group is precisely G. We exhibit these systems explicitly for many (low-rank) irreducible complex reflection groups in the Shephard-Todd classification.