Reflection groups, reflection arrangements, and invariant real varieties

TL;DR

This paper explores how real varieties invariant under reflection groups intersect flats of the associated reflection arrangements, generalizing Timofte's degree principle and providing bounds and evidence for various groups.

Contribution

It proves the conjecture for certain reflection groups, extends the degree principle, and offers bounds based on parabolic subgroup combinatorics.

Findings

Proved the conjecture for infinite types, rank ≤ 3, and F4 groups.

Provided computational evidence for H4.

Derived bounds on minimal flat dimensions meeting invariant varieties.

Abstract

Let $X$ be a nonempty real variety that is invariant under the action of a reflection group $G$. We conjecture that if $X$ is defined in terms of the first $k$ basic invariants of $G$ (ordered by degree), then $X$ meets a $k$-dimensional flat of the associated reflection arrangement. We prove this conjecture for the infinite types, reflection groups of rank at most $3$, and $F_4$ and we give computational evidence for $H_4$. 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 $X$ from the combinatorics of parabolic subgroups. We also give generalizations to real varieties invariant under Lie groups.