## 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:

 5e+99 Algebraic combinatorics
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