## 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.

Original language | English |
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Pages (from-to) | 1031-1045 |

Number of pages | 15 |

Journal | Proceedings of the American Mathematical Society |

Volume | 146 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2018 |

MoE publication type | A1 Journal article-refereed |

## Keywords

- Invariant real varieties
- Real orbit spaces
- Reflection arrangements
- Reflection groups