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Complex Hadamard Matrices contained in a Bose–Mesner algebra

Treść / Zawartość
Warianty tytułu
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.
Wydawca
Czasopismo
Rocznik
Tom
3
Numer
1
Opis fizyczny
Daty
otrzymano
2014-11-21
zaakceptowano
2015-04-22
online
2015-05-07
Twórcy
autor
  • Kobe Gakuin University, Kobe, 650-8586, Japan
  • 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]
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_1515_spma-2015-0009
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