Complex Reflection Groups as Differential Galois Groups

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.