Equiangular tight frames from complex Seidel matrices containing cube roots of unity

TL;DR

This paper characterizes complex Seidel matrices with cube roots of unity that have two eigenvalues, linking them to equiangular tight frames and directed graphs, and provides constructions for arbitrarily large frames.

Contribution

It introduces verifiable conditions for complex Seidel matrices with cube roots of unity to have two eigenvalues and connects these matrices to equiangular tight frames and directed graphs.

Findings

Characterization of complex Seidel matrices with two eigenvalues

Equivalence between such matrices and certain equiangular tight frames

Construction methods for large equiangular tight frames

Abstract

We derive easily verifiable conditions which characterize when complex Seidel matrices containing cube roots of unity have exactly two eigenvalues. The existence of such matrices is equivalent to the existence of equiangular tight frames for which the inner product between any two frame vectors is always a common multiple of the cube roots of unity. We also exhibit a relationship between these equiangular tight frames, complex Seidel matrices, and highly regular, directed graphs. We construct examples of such frames with arbitrarily many vectors.