Orthogonality as Graph with an Application to Finite Coxeter Groups

The common neighborhood of a set of nodes in a simple graph is a
Galois connection with associated closure operation on the power set
of the node set. An application of this concept to root systems
yields a new Galois connection on the (conjugacy classes of) parabolic
subgroups of a finite Coxeter group which we have used to refine
Howlett's description of the normalizers of its parabolic subgroups.
This talk is based on joint work G. Roehrle and J.M. Douglass
(arXiv:2509.15850).