# 2- and 3-Covariant Equiangular Tight Frames

## TL;DR

This paper investigates the symmetry properties of equiangular tight frames (ETFs), showing that certain classes are roux lines and exploring their group transitivity characteristics.

## Contribution

It characterizes the transitivity of ETFs' symmetry groups and proves that an infinite class of Gabor-Steiner ETFs are roux lines, extending understanding of their structure.

## Key findings

- Gabor-Steiner ETFs are roux lines.
- Symmetry groups of ETFs can be characterized in terms of transitivity.
- Infinite class of ETFs exhibit roux line structure.

## Abstract

Equiangular tight frames (ETFs) are configurations of vectors which are optimally geometrically spread apart and provide resolutions of the identity. Many known constructions of ETFs are group covariant, meaning they result from the action of a group on a vector, like all known constructions of symmetric, informationally complete, positive operator-valued measures. In this short article, some results characterizing the transitivity of the symmetry groups of ETFs will be presented as well as a proof that an infinite class of so-called Gabor-Steiner ETFs are roux lines, where roux lines are a generalization of doubly transitive lines.