Complex reflection groups as differential Galois groups
Carlos E. Arreche, Avery Bainbridge, Benjamin Obert, Alavi Ullah
TL;DR
The paper realizes complex reflection groups as differential Galois groups over the multivariate rational function field by leveraging invariant theory: it identifies $K=\mathbb{C}(z_1,\dots,z_n)$ with $L^G$, uses $\delta_i=\partial/\partial z_i$ to form a Δ-field, and constructs a PV extension with a concrete integrable system given by $A_i=\delta_i(J_{\boldsymbol{\phi}})J_{\boldsymbol{\phi}}^{-1}$; this yields $\mathrm{Gal}_{\Delta}(L/K)\simeq G$ in the natural representation. The approach extends prior Beukers–Heckman results to multivariate settings and provides an explicit algorithm to rewrite invariant expressions in the invariant coordinates, enabling concrete systems for primitive complex reflection groups of low rank. The paper demonstrates the method with a dihedral example and reports explicit outputs for the tetrahedral groups, highlighting both the practicality of the algorithm and the increasing complexity for higher ranks. This work advances explicit constructive realizations of complex reflection groups as differential Galois groups and offers a computational toolkit for the inverse differential Galois problem in multivariate contexts.
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.