Twisted extensions of spin models
A spin model of size \(n\) is understood as a matrix \(W= (W_{\text{a,b}})\) of degree \(n\) such that the entries \(W_{\text{a,b}}\) are non- zero complex numbers satisfying the following two equations \[ \begin{aligned} &\sum_{x=1}^n W_{\text{a},x}/ W_{\text{b},x}=0, \qquad \text{if a \(\neq\) b} \qquad \text{and} \tag{1}\\ &\sum_{x=1}^n W_{\text{a},x} W_{x,\text{b}}/ W_{\text{c},x}= \sqrt{n} W_{\text{a,b}}/ W_{\text{c,a}} W_{\text{c,b}}. \tag{2} \end{aligned} \] The main result of the author is the following extension theorem: Let \(W\) be a spin model of size \(n\), \(e= (e_1, \dots, e_n)\) a \((-1,1)\) \(n\)-tuple, and \(W'= (W_{\text{a,b}}')\) the matrix of degree \(n\) defined by \(W_{\text{a,b}}'= e_{\text a} e_{\text b} W_{\text{a,b}}\). Then the two matrices \[ \left( \begin{alignedat}{4} &W&\quad&W&\quad&W'&\quad-&W' \\ &W&\quad&W&\quad-&W'&\quad&W' \\ &W'&\quad-&W'&\quad&W&\quad&W \\ -&W'&\quad&W'&\quad&W&\quad&W \end{alignedat} \right) \qquad \text{and} \qquad \left( \begin{alignedat}{4} &W'&\quad-&W'&\quad&W&\quad&W \\ -&W'&\quad&W'&\quad&W&\quad&W \\ &W&\quad&W&\quad&W'&\quad-&W' \\ &W&\quad&W&\quad-&W'&\quad&W' \end{alignedat} \right) \] are spin models of size \(4n\). The proof is done by a skilful manipulation of equations similar to (1) and (2).
scientific article; zbMATH DE number 754561