Reflection groups, reflection arrangements, and invariant real varieties

2016/02/22 by Tobias Friedl, Cordian Riener, Friedl, Tobias +3
Computer Science · Mathematics · #14P05 #14P10 #20F55 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1602.06732

openalex publication_date 2016/02/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 F₄ and we give computational evidence for H₄. 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.

Related

• A geometrical construction for the polynomial invariants of some reflection groups
• Fundamental theorems for invariants of a class of pseudo-reflection groups
• Freeness of multi-reflection arrangements via primitive vector fields
• Reflection groups in algebraic geometry
• A Geometric Interpretation of the Characteristic Polynomial of Reflection Arrangements