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On Normalizers of Parabolic Subgroups of Quaternionic Reflection Groups

Gerhard Roehrle, Johannes Schmitt

Abstract

By work of Howlett and Muraleedaran--Taylor, a parabolic subgroup of a real or complex reflection group always admits a complement in its normalizer. In this note, we investigate this phenomenon for quaternionic reflection groups. Here, in contrast to the real and complex setting, we find that complements of parabolic subgroups do not exist in general. Indeed, there are infinitely many examples of quaternionic reflection groups in arbitrary rank greater than 2 with a parabolic subgroup that does not admit a complement in its normalizer. We give a full classification of parabolic subgroups of irreducible quaternionic reflection groups and describe their complements, if the latter exist.

On Normalizers of Parabolic Subgroups of Quaternionic Reflection Groups

Abstract

By work of Howlett and Muraleedaran--Taylor, a parabolic subgroup of a real or complex reflection group always admits a complement in its normalizer. In this note, we investigate this phenomenon for quaternionic reflection groups. Here, in contrast to the real and complex setting, we find that complements of parabolic subgroups do not exist in general. Indeed, there are infinitely many examples of quaternionic reflection groups in arbitrary rank greater than 2 with a parabolic subgroup that does not admit a complement in its normalizer. We give a full classification of parabolic subgroups of irreducible quaternionic reflection groups and describe their complements, if the latter exist.

Paper Structure

This paper contains 17 sections, 24 theorems, 48 equations, 8 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Parabolic subgroups
  4. Reduction to the irreducible case
  5. Complex reflection groups
  6. The classification of quaternionic reflection groups
  7. The Kleinian groups
  8. Imprimitive quaternionic reflection groups
  9. Complements of parabolic subgroups
  10. The exceptional imprimitive groups in rank 2
  11. Primitive groups with imprimitive complexification
  12. The reflections
  13. The parabolic subgroups
  14. Normalizers of parabolic subgroups
  15. Primitive groups with primitive complexification
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $G$ be an irreducible quaternionic reflection group and let $P\leq G$ be a parabolic subgroup. Then $P$ has a complement in its normalizer $N_G(P)$ except if the pair $(G, P)$ is one of the following: $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (54)

  • Theorem 1.1
  • Remark 1.2
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.3
  • proof
  • Proposition 3.5
  • ...and 44 more