Bose-Mesner Algebras Associated with Four-Weight Spin Models
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- Bannai, E. Graphs Comb (2001) 17: 589. doi:10.1007/PL00007251
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It is known that for each matrix Wi and it's transpose tWi in any four-weight spin model (X, W1, W2, W3, W4; D), there is attached the Bose-Mesner algebra of an association scheme, which we call Nomura algebra. They are denoted by N(Wi) and N(tWi) = N′(Wi) respectively. H. Guo and T. Huang showed that some of them coincide with a self-dual Bose-Mesner algebra, that is, N(W1) = N′(W1) = N(W3) = N′(W3) holds. In this paper we show that all of them coincide, that is, N(Wi), N′(Wi), i=1, 2, 3, 4, are the same self-dual Bose-Mesner algebra.