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Generalized generalized spin models (four-weight spin models). (English) Zbl 0848.05072

Summary: The concept of spin model was introduced by V. F. R. Jones. Kawagoe, Munemasa and Watatani generalized it by dropping the symmetric condition, and defined a generalized spin model. In this paper, by further generalizing the concept using four functions, we define a generalized generalized spin model (four-weight spin model). Namely, \((X, w_1, w_2, w_3, w_4)\) is a generalized generalized spin model (four-weight spin model), if \(X\) is a finite set and \(w_i\) \((i = 1, 2, 3, 4)\) are complex valued functions on \(X \times X\) satisfying the following conditions: \[ w_1 (\alpha, \beta) w_3 (\beta, \alpha) = 1, \quad w_2 (\alpha, \beta) w_4 (\beta, \alpha) = 1 \tag{1} \] for any \(\alpha, \beta\) in \(X\), \[ \sum_{x \in X} w_1 (\alpha, x) w_3 (x, \beta) = n \delta_{\alpha, \beta}, \quad \sum_{x \in X} w_2 (\alpha,x) w_4 (x, \beta) = n \delta_{\alpha, \beta} \tag{2} \] for any \(\alpha\) and \(\beta\) in \(X\), \[ \sum_{x \in X} w_1 (\alpha, x) w_1 (x, \beta) w_4 (\gamma, x) = Dw_1 (\alpha, \beta) w_4 (\gamma, \alpha) w_4 (\gamma, \beta) \tag{3a} \] and \[ \sum_{x \in X} w_1 (x, \alpha) w_1 (\beta, x) w_4 (x, \gamma) = Dw_1 (\beta, \alpha) w_4 (\alpha, \gamma) w_4 (\beta, \gamma) \tag{3b} \] for any \(\alpha, \beta\) and \(\gamma\) in \(X\), where \(D^2 = n = |X |\).
We call as generalized spin models (two-weight spin models), the special cases of generalized generalized spin models (four-weight spin models), where there are only two functions \(w_+\) and \(w_-\) from \(X \times X\) to \(\mathbb{C}\) with two of \(w_1, w_2, w_3, w_4\) being in \(\{w_+, ^tw_+\}\) and the remaining two of \(w_1, w_2, w_3, w_4\) being in \(\{w_-, ^tw_-\}\). We see that we have three types of generalized spin models (two-weight spin models), namely Jones type, pseudo-Jones type, and Hadamard type. We also see that Kawagoe-Munemasa-Watatani’s generalized spin model is one special case of Jones type, and Jones’ original spin model is a further special case of it. Here we emphasize that there are actually interesting spin models which are considerably different from the original concept of spin model defined by Jones.

MSC:

05E99 Algebraic combinatorics
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