Differential Structures on Braid Hyperplane Arrangements

Abstract

This paper establishes an explicit computational bridge between the algebraic freeness of braid arrangements and the quantum integrability of the Calogero–Moser system. While the foundational properties of logarithmic derivations and Dunkl operators are well established, their direct application to constructing physical integrals of motion is often left abstract. We present a constructive framework that utilises the symmetric group invariant derivation basis of the arrangement to explicitly generate the commuting differential operators required for integrability. By applying this methodology to the 3-particle system (A2 arrangement), we demonstrate how the purely algebraic exponents of the arrangement directly dictate the existence and degrees of conserved physical quantities. This explicitly connects the discrete combinatorics of reflection arrangements with the continuous dynamics of exactly solvable quantum models.