Seidel's legacy and the existence of complex equiangular Parseval frames

Seidel's combinatorial approach to the construction of real, symmetric matrices with unimodular entries and two eigenvalues has produced many equiangular Parseval frames for real Hilbert spaces. We follow Seidel's footsteps and develop a corresponding combinatorial characterization of complex Seidel matrices belonging to equiangular Parseval frames. We deduce necessary conditions for the existence of complex Seidel matrices containing pth roots of unity and having exactly two eigenvalues, under the assumption that p is prime. Explicitly examining the necessary conditions for p = 5, for example, rules out the existence of many such frames with a number of vectors less than 50. Nevertheless, there are examples, which we confirm by constructing p² × p² Seidel matrices containing pth roots of unity and having two eigenvalues. and thus the existence of the associated complex equiangular Parseval frames, for any p ¿ 2.