Complex Hadamard Matrices contained in a Bose–Mesner algebra

• Autorzy: Takuya Ikuta , Akihiro Munemasa
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

association scheme; complex Hadamard matrix; type-II matrix

Kategorie tematyczne

• 05E30: Association schemes, strongly regular graphs

• Wydawca: De Gruyter Open
• Czasopismo: Special Matrices
• Rocznik: 2015
• Tom: 3
• Numer: 1

Daty

otrzymano

2014-11-21

zaakceptowano

2015-04-22

online

2015-05-07

Twórcy

autor

Takuya Ikuta

• Kobe Gakuin University, Kobe, 650-8586, Japan

autor

Akihiro Munemasa

• Tohoku University, Sendai, 980-8579, Japan,

Bibliografia

• [1] E. Bannai and T. Ito, Algebraic Combinatorics I: Association Schemes, Benjamin/Cummings, Menlo Park, 1984.
• [2] W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997),235–265.
• [3] A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer-Verlag, Berlin, Heidelberg, 1989.
• [4] A. Chan, Complex Hadamard matrices and strongly regular graphs, arXiv:1102.5601.
• [5] A. Chan and C. Godsil, Type-II matrices and combinatorial structures, Combinatorica, 30 (2010), 1–24.[WoS][Crossref]
• [6] A. Chan and R. Hosoya, Type-II matrices attached to conference graphs, J. Algebraic Combin. 20 (2004), 341–351.
• [7] R. Craigen, Equivalence classes of inverse orthogonal and unit Hadamard matrices, Bull. Austral. Math. Soc. 44 (1991), no. 1,109–115.[Crossref]
• [8] E. van Dam, Three-class association schemes, J. Algebraic Combin. 10 (1999), 69–107.
• [9] J. M. Goethals and J. J. Seidel, Strongly regular graphs derived from combinatorial designs, Can. J.Math., 22, (1970), 597–614.
• [10] U. Haagerup, Orthogonal maximal Abelian *-subalgebras of n × n matrices and cyclic n-roots, Operator Algebras and QuantumField Theory (Rome), Cambridge, MA, International Press, (1996), 296–322.
• [11] R. Hosoya and H. Suzuki, Type II matrices and their Bose-Mesner algebras, J. Algebraic Combin. 17 (2003), 19–37.
• [12] F. Jaeger, M. Matsumoto, and K. Nomura, Bose-Mesner algebras related to type II matrices and spin models, J. AlgebraicCombin. 8 (1998), 39–72.
• [13] R. Nicoara, A finiteness result for commuting squares of matrix algebras, J. Operator Theory 55 (2006), 101–116.
• [14] K. Nomura, Type II matrices of size five, Graphs Combin.15 (1999), 79–92.
• [15] A. D. Sankey, Type-II matrices in weighted Bose-Mesner algebras of ranks 2 and 3, J. Algebraic Combin. 32 (2010), 133–153.
• [16] F. Szöllősi, Exotic complex Hadamard matrices and their equivalence, Cryptogr. Commun. 2 (2010), no. 2, 187–198.
• [17] W. Tadej and K. Życzkowski A concise guide to complex Hadamard matrices, Open Syst. Inf. Dyn. 13 (2006), 133–177.[Crossref]

Identyfikatory

DOI

10.1515/spma-2015-0009