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The quaternionic systems of imprimitivity for the reflection groups of rank two

Shayne Waldron

TL;DR

The work develops a practical framework to determine all systems of imprimitivity for irreducible rank-two reflection groups over $\mathbb{R},\mathbb{C},\mathbb{H}$ by parameterizing orthogonal bases via $(1,q)$ and enforcing monomial representations through two key equations. It yields explicit classifications for real, complex, and quaternionic cases, showing, for example, that $D_4$ is the unique imprimitive real rank-two group and that only $G_{12},G_{13},G_{22}$ carry quaternionic imprimitivity among primitive complex groups. The results reveal that, generally, larger groups have fewer imprimitivity systems, with a precise taxonomy that includes infinite quaternionic families and a maximal finite count of 10 in certain quaternionic scenarios. Overall, the paper unifies disparate strands of the literature on reflection groups by providing a coherent, computation-driven approach to imprimitivity across division algebras and clarifies how conjugacy and group structure govern the existence and multiplicity of these systems.

Abstract

Given an explicit presentation of a reflection group of rank two (or any rank two group for that matter), we give a simple procedure for calculating all its systems of imprimitivity, when viewed as a matrix group over the quaternions. This is applied to all the reflection groups, in particular the quaternionic reflection groups, thereby unifying a number of results and ideas in the literature. For example, a primitive complex reflection group of rank two has either uncountably many quaternionic systems of imprimitivity (3 cases) or none (16 cases).

The quaternionic systems of imprimitivity for the reflection groups of rank two

TL;DR

The work develops a practical framework to determine all systems of imprimitivity for irreducible rank-two reflection groups over by parameterizing orthogonal bases via and enforcing monomial representations through two key equations. It yields explicit classifications for real, complex, and quaternionic cases, showing, for example, that is the unique imprimitive real rank-two group and that only carry quaternionic imprimitivity among primitive complex groups. The results reveal that, generally, larger groups have fewer imprimitivity systems, with a precise taxonomy that includes infinite quaternionic families and a maximal finite count of 10 in certain quaternionic scenarios. Overall, the paper unifies disparate strands of the literature on reflection groups by providing a coherent, computation-driven approach to imprimitivity across division algebras and clarifies how conjugacy and group structure govern the existence and multiplicity of these systems.

Abstract

Given an explicit presentation of a reflection group of rank two (or any rank two group for that matter), we give a simple procedure for calculating all its systems of imprimitivity, when viewed as a matrix group over the quaternions. This is applied to all the reflection groups, in particular the quaternionic reflection groups, thereby unifying a number of results and ideas in the literature. For example, a primitive complex reflection group of rank two has either uncountably many quaternionic systems of imprimitivity (3 cases) or none (16 cases).
Paper Structure (6 sections, 11 theorems, 134 equations, 2 figures, 5 tables)

This paper contains 6 sections, 11 theorems, 134 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. How to find systems of imprimitivity
  3. The systems of imprimitivity of the real reflection groups
  4. The complex reflection groups and their systems of imprimitivity
  5. The systems of imprimitivity of the quaternionic reflection groups
  6. Concluding remarks

Key Result

Proposition 2.1

Every orthogonal system for $\mathbb{F}^2$ is given by a vector with different values of $q$ giving different orthogonal systems, unless $|q|=1$, in which case the same orthogonal system is given by $(1,\pm q)$ (i.e., these are counted twice above).

Figures (2)

  • Figure 1: The inclusions between the primitive complex reflection groups $G_4,\ldots,G_{22}$. Those which turn out to have quaternionic systems of imprimitivity are shaded.
  • Figure 2: The inclusions of the reflection groups for ${\cal T},{\cal O},{\cal I}$ given in Table \ref{['TOIDngroups-table']}, with those having additional systems of imprimitivity shaded in grey.

Theorems & Definitions (22)

  • Proposition 2.1
  • Lemma 2.1
  • Theorem 3.1
  • Example 3.1
  • Lemma 3.1
  • Theorem 4.1
  • Example 4.1
  • Example 4.2
  • Lemma 4.1
  • Example 4.3
  • ...and 12 more