7 research outputs found
Complex Hadamard matrices contained in a Bose-Mesner algebra
A complex Hadamard matrix is a square matrix H with complex entries of
absolute value 1 satisfying , where stands for the Hermitian
transpose and I is the identity matrix of order . In this paper, we first
determine the image of a certain rational map from the -dimensional complex
projective space to . Applying this result with ,
we give constructions of complex Hadamard matrices, and more generally, type-II
matrices, in the Bose-Mesner algebra of a certain 3-class symmetric association
scheme. In particular, we recover the complex Hadamard matrices of order 15
found by Ada Chan. We compute the Haagerup sets to show inequivalence of
resulting type-II matrices, and determine the Nomura algebras to show that the
resulting matrices are not decomposable into generalized tensor products.Comment: 28 pages + Appendix A + Appendix