Optimal line packings from nonabelian groups

TL;DR

This paper constructs optimal line packings using nonabelian group schemes, introduces a new integrality condition for equiangular frames, and presents the first infinite family of such packings from nonabelian groups.

Contribution

It introduces a novel method for constructing optimal line packings via nonabelian groups and provides the first infinite family of equiangular tight frames from these groups.

Findings

Constructed optimal line packings using group schemes.

Derived a necessary integrality condition for equiangular frames.

Presented an infinite family of such packings from Suzuki 2-groups.

Abstract

We use group schemes to construct optimal packings of lines through the origin. In this setting, optimal line packings are naturally characterized using representation theory, which in turn leads to a necessary integrality condition for the existence of equiangular central group frames. We conclude with an infinite family of optimal line packings using the group schemes associated with certain Suzuki 2-groups, specifically, extensions of Heisenberg groups. Notably, this is the first known infinite family of equiangular tight frames generated by representations of nonabelian groups.