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Tytuł artykułu
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Warianty tytułu
Języki publikacji
Abstrakty
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
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Opis fizyczny
Daty
otrzymano
2014-11-21
zaakceptowano
2015-04-22
online
2015-05-07
Twórcy
autor
- Kobe Gakuin University, Kobe, 650-8586, Japan
autor
- 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 Quantum Field 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. Algebraic Combin. 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