On Normalizers of Parabolic Subgroups of Quaternionic Reflection Groups

TL;DR

This paper explores the existence of complements of parabolic subgroups in quaternionic reflection groups, revealing that unlike real and complex cases, such complements often do not exist, and provides a full classification.

Contribution

It provides the first comprehensive classification of parabolic subgroups in quaternionic reflection groups and characterizes when complements exist.

Findings

Complements of parabolic subgroups do not always exist in quaternionic reflection groups.

There are infinitely many quaternionic reflection groups with non-complemented parabolic subgroups.

A full classification of parabolic subgroups and their complements in irreducible quaternionic reflection groups is provided.

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.