##
**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.

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 |