Complex Hadamard matrices contained in a Bose-Mesner algebra

Abstract

A complex Hadamard matrix is a square matrix H with complex entries of absolute value 1 satisfying $H{H}^{\ast }=nI$, where $\ast$ stands for the Hermitian transpose and I is the identity matrix of order $n$. In this paper, we first determine the image of a certain rational map from the $d$-dimensional complex projective space to ${\mathbb{C}}^{d(d+1)/2}$. Applying this result with $d=3$, 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.

28 pages + Appendix A + Appendix B