The geometry of the six quaternionic equiangular lines in $\mathbb{H}^2$

TL;DR

This paper explores the structure of six quaternionic equiangular lines in H^2, presenting their geometric properties as an orbit of a specific quaternionic reflection group and connecting them to optimal spherical designs.

Contribution

It provides a simple presentation of the six quaternionic equiangular lines as an orbit of a primitive quaternionic reflection group and links this to other optimal line packings and designs.

Findings

Six quaternionic equiangular lines form an orbit of a specific reflection group.

Other orbits yield optimal spherical designs with different line counts and angles.

Connections to classical collineation groups and methods for finding quaternionic line systems.

Abstract

We give a simple presentation of the six quaternionic equiangular lines in $\mathbb{H}^2$ as an orbit of the primitive quaternionic reflection group of order 720 (which is isomorphic to 2.A_6 the double cover of $A_6)$. Other orbits of this group are also seen to give optimal spherical designs (packings) of 10, 15 and 20 lines in $\mathbb{H}^2$, with angles { 1/3, 2/3 }, { 1/4, 5/8 } and { 0, 1/3, 2/3 }, respectively. We consider the origins of this reflection group as one of Blichfeldt's "finite collineation groups" for lines in $\mathbb{C}^4$, and general methods for finding nice systems of quaternionic lines.