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.

翻译：复反射群是半单李代数的Weyl群的推广，更一般地也是有限Coxeter群的推广。自Shephard与Todd于20世纪50年代引入并完成其完全分类以来，因其在组合数学、表示论、纽结理论和数学物理等诸多领域的应用而得到深入研究。针对任意给定的复反射群G，我们提出一种新方法，用于构造可积线性微分方程组，其微分Galois群恰好为G。我们在Shephard-Todd分类中对许多（低秩）不可约复反射群给出了这些方程组的显式构造。