Reflection groups, reflection arrangements, and invariant real varieties

Tobias Friedl, Cordian Riener, Raman Sanyal

Research output: Contribution to journalArticleScientificpeer-review

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.

Original languageEnglish
Pages (from-to)1031-1045
Number of pages15
JournalProceedings of the American Mathematical Society
Volume146
Issue number3
DOIs
Publication statusPublished - 2018
MoE publication typeA1 Journal article-refereed

Keywords

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

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