Complex equiangular Parseval frames and Seidel matrices containing 𝑝th roots of unity

Abstract

We derive necessary conditions for the existence of complex Seidel matrices containing p p th roots of unity and having exactly two eigenvalues, under the assumption that p p is prime. The existence of such matrices is equivalent to the existence of equiangular Parseval frames with Gram matrices whose off-diagonal entries are a common multiple of the p p th roots of unity. Explicitly examining the necessary conditions for p = 5 p=5 and p = 7 p=7 rules out the existence of many such frames with a number of vectors less than 50, similar to previous results in the cube roots case. On the other hand, we confirm the existence of p 2 × p 2 p^2\times p^2 Seidel matrices containing p p th roots of unity, and thus the existence of the associated complex equiangular Parseval frames, for any p ≥ 2 p\ge 2 . The construction of these Seidel matrices also yields a family of previously unknown Butson-type complex Hadamard matrices.