Reflection groups, reflection arrangements, and invariant real varieties

Abstract

Let X X be a nonempty real variety that is invariant under the action of a reflection group G G . We conjecture that if X X is defined in terms of the first k k basic invariants of G G (ordered by degree), then X X meets a k k -dimensional flat of the associated reflection arrangement. We prove this conjecture for the infinite types, reflection groups of rank at most 3 3 , and F 4 F_4 and we give computational evidence for H 4 H_4 . This is a generalization of Timofte’s degree principle to reflection groups. For general reflection groups, we compute nontrivial upper bounds on the minimal dimension of flats of the reflection arrangement meeting X X from the combinatorics of parabolic subgroups. We also give generalizations to real varieties invariant under Lie groups.