Complex Hadamard Matrices contained in a Bose-Mesner algebra

Takuya Ikuta; Akihiro Munemasa

doi:10.1515/spma-2015-0009

Complex Hadamard Matrices contained in a Bose-Mesner algebra

Takuya Ikuta, Akihiro Munemasa

INF - Computer and Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

Abstract

Acomplex Hadamard matrix is a square matrix H with complex entries of absolute value 1 satisfying HH∗= nI, where∗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 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.

Original language	English
Pages (from-to)	91-110
Number of pages	20
Journal	Special Matrices
Volume	3
Issue number	1
DOIs	https://doi.org/10.1515/spma-2015-0009
Publication status	Published - 2015 Jan

Keywords

Association scheme
Complex Hadamard matrix
Type-II matrix

ASJC Scopus subject areas

Algebra and Number Theory
Geometry and Topology

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title = "Complex Hadamard Matrices contained in a Bose-Mesner algebra",

abstract = "Acomplex Hadamard matrix is a square matrix H with complex entries of absolute value 1 satisfying HH∗= nI, where∗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 Cd(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.",

keywords = "Association scheme, Complex Hadamard matrix, Type-II matrix",

author = "Takuya Ikuta and Akihiro Munemasa",

note = "Funding Information: The work of T.I. was supported by JSPS KAKENHI grant number 25400215, and that of A.M. was supported by JSPS KAKENHI grant number 26400003. Publisher Copyright: {\textcopyright} 2015 Takuya Ikuta et al., licensee De Gruyter Open.",

year = "2015",

month = jan,

doi = "10.1515/spma-2015-0009",

language = "English",

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AU - Ikuta, Takuya

AU - Munemasa, Akihiro

N1 - Funding Information: The work of T.I. was supported by JSPS KAKENHI grant number 25400215, and that of A.M. was supported by JSPS KAKENHI grant number 26400003. Publisher Copyright: © 2015 Takuya Ikuta et al., licensee De Gruyter Open.

PY - 2015/1

Y1 - 2015/1

N2 - Acomplex Hadamard matrix is a square matrix H with complex entries of absolute value 1 satisfying HH∗= nI, where∗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 Cd(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.

AB - Acomplex Hadamard matrix is a square matrix H with complex entries of absolute value 1 satisfying HH∗= nI, where∗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 Cd(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.

KW - Association scheme

KW - Complex Hadamard matrix

KW - Type-II matrix

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JO - Special Matrices

JF - Special Matrices

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ER -

Complex Hadamard Matrices contained in a Bose-Mesner algebra

Abstract

Keywords

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