# Combinatorics Seminar 2015

Date: Wed, Oct 21, 2015

Time: 15:00 - 15:50

Venue: Middle Meeting Room

Title: 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 $HH^\ast= nI$, where $\ast$ stands for the Hermitian transpose and $I$ is the identity matrix of order $n$. They are the natural generalization of real Hadamard matrices. A type-II matrix, or an inverse orthogonal matrix, is a square matrix $W$ with nonzero complex entries satisfying $W{W^{(-)}}^\top=nI$, where $W^{(-)}$ denotes the entrywise inverse of $W$. Obviously, a complex Hadamard matrix is a type-II matrix.

In this talk, 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. Moreover, we give constructions of complex Hadamard matrices in the Bose-Mesner algebra of a certain $4$-class symmetric association scheme.

This is based on joint work with Akihiro Munemasa.

Slides: View slides

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