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Systems of imprimitivity for rank two quaternionic reflection groups

Donald E Taylor

TL;DR

This work resolves omissions in the list of imprimitive rank-two quaternionic reflection groups by classifying groups G(K,H,φ) that act irreducibly and imprimitive on a 2-dimensional quaternionic space, and by determining when multiple systems of imprimitivity occur. It provides explicit constructions, automorphism-based criteria, and generators for the cyclic, binary dihedral, and binary polyhedral families, as well as primitive groups with imprimitive complexification. A key outcome is the demonstration that certain primitive complex reflection groups, notably the ST(12), ST(13), and ST(22) types, have infinitely many systems of imprimitivity in the quaternionic setting, and many isomorphisms between complex-type and quaternionic-type realizations are established. The paper also furnishes a comprehensive revised table of imprimitive irreducible rank-two quaternionic reflection groups and clarifies conjugacy relations across all major families, with implications for related areas such as the McKay correspondence and symplectic resolutions.

Abstract

We revise the enumeration of the imprimitive rank two quaternionic reflection groups, adding missing groups and establishing isomorphisms between groups in the published tables. The isomorphisms are obtained as a consequence of the determination of the reflection groups with more than one system of imprimitivity. We find that there are primitive complex reflection groups which have infinitely many systems of imprimitivity when represented as quaternionic reflection groups.

Systems of imprimitivity for rank two quaternionic reflection groups

TL;DR

This work resolves omissions in the list of imprimitive rank-two quaternionic reflection groups by classifying groups G(K,H,φ) that act irreducibly and imprimitive on a 2-dimensional quaternionic space, and by determining when multiple systems of imprimitivity occur. It provides explicit constructions, automorphism-based criteria, and generators for the cyclic, binary dihedral, and binary polyhedral families, as well as primitive groups with imprimitive complexification. A key outcome is the demonstration that certain primitive complex reflection groups, notably the ST(12), ST(13), and ST(22) types, have infinitely many systems of imprimitivity in the quaternionic setting, and many isomorphisms between complex-type and quaternionic-type realizations are established. The paper also furnishes a comprehensive revised table of imprimitive irreducible rank-two quaternionic reflection groups and clarifies conjugacy relations across all major families, with implications for related areas such as the McKay correspondence and symplectic resolutions.

Abstract

We revise the enumeration of the imprimitive rank two quaternionic reflection groups, adding missing groups and establishing isomorphisms between groups in the published tables. The isomorphisms are obtained as a consequence of the determination of the reflection groups with more than one system of imprimitivity. We find that there are primitive complex reflection groups which have infinitely many systems of imprimitivity when represented as quaternionic reflection groups.
Paper Structure (16 sections, 27 theorems, 29 equations, 5 tables)

This paper contains 16 sections, 27 theorems, 29 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. The finite subgroups of $\mathbb{S}^3$
  4. The irreducible imprimitive groups of rank two
  5. The cyclic family.
  6. The binary dihedral family.
  7. The families of the binary polyhedral groups $\mathcal{T}$, $\mathcal{O}$ and $\mathcal{I}$.
  8. Primitive groups with imprimitive complexification
  9. Systems of imprimitivity
  10. The binary dihedral groups.
  11. The binary tetra-, octa- and icosahedral groups.
  12. The extended binary polyhedral groups.
  13. Conjugacy
  14. Conjugacy in the binary tetra-, octa- and icosahedral families.
  15. Conjugacy of groups of the binary dihedral family
  16. ...and 1 more sections

Key Result

Lemma 4.2

$G(K,H,\varphi)$ is a quaternionic reflection group if and only if $L_\varphi$ generates $K$.

Theorems & Definitions (62)

  • Definition 2.1
  • Definition 4.1
  • Lemma 4.2
  • proof
  • Example 4.3
  • Definition 4.4
  • Lemma 4.5: cohen:1980
  • Corollary 4.6
  • Theorem 4.7: cohen:1980
  • proof
  • ...and 52 more