Equiangular Frames and Signature Sets

TL;DR

This paper explores the connection between real and complex equiangular frames and special subsets within groups, introducing signature and quasi-signature sets to generate and analyze these frames.

Contribution

It introduces the concepts of signature and quasi-signature sets and demonstrates their role in constructing real and complex equiangular frames from groups.

Findings

Many equiangular frames can be derived from signature sets in groups.

Extension of results to complex equiangular frames with cube root of unity inner products.

Identification of group-based methods for constructing equiangular frames.

Abstract

We will present a relation between real equiangular frames and certain special sets in groups which we call signature sets and show that many equiangular frames arise in this manner. Then we will define quasi-signature sets and will examine equiangular frames associated to these subsets of groups. We will extend these results to complex equiangular frames where the inner product between any pair of vectors is a common multiple of a cube root of unity and exhibit equiangular frames that arise from groups in this manner.