Magnitude homology of tope graphs

Abstract

We completely determine the magnitude homology of tope graphs of real hyperplane arrangements. Their ranks can be described as the Hilbert functions of the Stanley--Reisner rings of certain simplicial complexes naturally associated with the arrangements. For Coxeter arrangements, this gives a computation of the magnitude homology of the Cayley graph of the corresponding Coxeter group. We also prove the homological reciprocity for central arrangements conjectured by Koizumi--Liu. The proof combines poset combinatorics, the Edelman--Walker theorem, and Alexander duality.